2引数が同じ3引数の論理演算子
2引数が同じ3引数の論理演算子
\(P,Q\)は命題変数とする。
肯定と肯定パターン
肯定と否定パターン
\(P,Q\)は命題変数とする。
肯定と肯定パターン
(1)
\[ P\lor\left(P\lor Q\right)\Leftrightarrow P\lor Q \](2)
\[ P\lor\left(P\land Q\right)\Leftrightarrow P \](3)
\[ P\lor\left(P\rightarrow Q\right)\Leftrightarrow1 \](4)
\[ P\lor\left(P\leftarrow Q\right)\Leftrightarrow P\leftarrow Q \](5)
\[ P\lor\left(P\leftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](6)
\[ P\lor\left(P\downarrow Q\right)\Leftrightarrow P\leftarrow Q \](7)
\[ P\lor\left(P\uparrow Q\right)\Leftrightarrow1 \](8)
\[ P\lor\left(P\nrightarrow Q\right)\Leftrightarrow P \](9)
\[ P\lor\left(P\nleftarrow Q\right)\Leftrightarrow P\lor Q \](10)
\[ P\lor\left(P\nleftrightarrow Q\right)\Leftrightarrow P\lor Q \](11)
\[ P\land\left(P\lor Q\right)\Leftrightarrow P \](12)
\[ P\land\left(P\land Q\right)\Leftrightarrow P\land Q \](13)
\[ P\land\left(P\rightarrow Q\right)\Leftrightarrow P\land Q \](14)
\[ P\land\left(P\leftarrow Q\right)\Leftrightarrow P \](15)
\[ P\land\left(P\leftrightarrow Q\right)\Leftrightarrow P\land Q \](16)
\[ P\land\left(P\downarrow Q\right)\Leftrightarrow0 \](17)
\[ P\land\left(P\uparrow Q\right)\Leftrightarrow P\nrightarrow Q \](18)
\[ P\land\left(P\nrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](19)
\[ P\land\left(P\nleftarrow Q\right)\Leftrightarrow0 \](20)
\[ P\land\left(P\nleftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](21)
\[ P\rightarrow\left(P\lor Q\right)\Leftrightarrow1 \](22)
\[ P\rightarrow\left(P\land Q\right)\Leftrightarrow P\rightarrow Q \](23)
\[ P\rightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\rightarrow Q \](24)
\[ P\rightarrow\left(P\leftarrow Q\right)\Leftrightarrow1 \](25)
\[ P\rightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \](26)
\[ P\rightarrow\left(P\downarrow Q\right)\Leftrightarrow\lnot P \](27)
\[ P\rightarrow\left(P\uparrow Q\right)\Leftrightarrow P\uparrow Q \](28)
\[ P\rightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\uparrow Q \](29)
\[ P\rightarrow\left(P\nleftarrow Q\right)\Leftrightarrow\lnot P \](30)
\[ P\rightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\uparrow Q \](31)
\[ P\leftarrow\left(P\lor Q\right)\Leftrightarrow P\leftarrow Q \](32)
\[ P\leftarrow\left(P\land Q\right)\Leftrightarrow1 \](33)
\[ P\leftarrow\left(P\rightarrow Q\right)\Leftrightarrow P \](34)
\[ P\leftarrow\left(P\leftarrow Q\right)\Leftrightarrow P\lor Q \](35)
\[ P\leftarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\lor Q \](36)
\[ P\leftarrow\left(P\downarrow Q\right)\Leftrightarrow P\lor Q \](37)
\[ P\leftarrow\left(P\uparrow Q\right)\Leftrightarrow P \](38)
\[ P\leftarrow\left(P\nrightarrow Q\right)\Leftrightarrow1 \](39)
\[ P\leftarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\leftarrow Q \](40)
\[ P\leftarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](41)
\[ P\leftrightarrow\left(P\lor Q\right)\Leftrightarrow P\leftarrow Q \](42)
\[ P\leftrightarrow\left(P\land Q\right)\Leftrightarrow P\rightarrow Q \](43)
\[ P\leftrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\land Q \](44)
\[ P\leftrightarrow\left(P\leftarrow Q\right)\Leftrightarrow P\lor Q \](45)
\[ P\leftrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow Q \](46)
\[ P\leftrightarrow\left(P\downarrow Q\right)\Leftrightarrow P\nleftarrow Q \](47)
\[ P\leftrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\nrightarrow Q \](48)
\[ P\leftrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\uparrow Q \](49)
\[ P\leftrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\downarrow Q \](50)
\[ P\leftrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow\lnot Q \](51)
\[ P\downarrow\left(P\lor Q\right)\Leftrightarrow P\downarrow Q \](52)
\[ P\downarrow\left(P\land Q\right)\Leftrightarrow\lnot P \](53)
\[ P\downarrow\left(P\rightarrow Q\right)\Leftrightarrow0 \](54)
\[ P\downarrow\left(P\leftarrow Q\right)\Leftrightarrow P\nleftarrow Q \](55)
\[ P\downarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](56)
\[ P\downarrow\left(P\downarrow Q\right)\Leftrightarrow P\nleftarrow Q \](57)
\[ P\downarrow\left(P\uparrow Q\right)\Leftrightarrow0 \](58)
\[ P\downarrow\left(P\nrightarrow Q\right)\Leftrightarrow\lnot P \](59)
\[ P\downarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\downarrow Q \](60)
\[ P\downarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\downarrow Q \](61)
\[ P\uparrow\left(P\lor Q\right)\Leftrightarrow\lnot P \](62)
\[ P\uparrow\left(P\land Q\right)\Leftrightarrow P\uparrow Q \](63)
\[ P\uparrow\left(P\rightarrow Q\right)\Leftrightarrow P\uparrow Q \](64)
\[ P\uparrow\left(P\leftarrow Q\right)\Leftrightarrow\lnot P \](65)
\[ P\uparrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\uparrow Q \](66)
\[ P\uparrow\left(P\downarrow Q\right)\Leftrightarrow1 \](67)
\[ P\uparrow\left(P\uparrow Q\right)\Leftrightarrow P\rightarrow Q \](68)
\[ P\uparrow\left(P\nrightarrow Q\right)\Leftrightarrow P\rightarrow Q \](69)
\[ P\uparrow\left(P\nleftarrow Q\right)\Leftrightarrow1 \](70)
\[ P\uparrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \](71)
\[ P\nrightarrow\left(P\lor Q\right)\Leftrightarrow0 \](72)
\[ P\nrightarrow\left(P\land Q\right)\Leftrightarrow P\nrightarrow Q \](73)
\[ P\nrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](74)
\[ P\nrightarrow\left(P\leftarrow Q\right)\Leftrightarrow0 \](75)
\[ P\nrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](76)
\[ P\nrightarrow\left(P\downarrow Q\right)\Leftrightarrow P \](77)
\[ P\nrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\land Q \](78)
\[ P\nrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\land Q \](79)
\[ P\nrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P \](80)
\[ P\nrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\land Q \](81)
\[ P\nleftarrow\left(P\lor Q\right)\Leftrightarrow P\nleftarrow Q \](82)
\[ P\nleftarrow\left(P\land Q\right)\Leftrightarrow0 \](83)
\[ P\nleftarrow\left(P\rightarrow Q\right)\Leftrightarrow\lnot P \](84)
\[ P\nleftarrow\left(P\leftarrow Q\right)\Leftrightarrow P\downarrow Q \](85)
\[ P\nleftarrow\left(P\leftrightarrow Q\right)\Leftrightarrow P\downarrow Q \](86)
\[ P\nleftarrow\left(P\downarrow Q\right)\Leftrightarrow P\downarrow Q \](87)
\[ P\nleftarrow\left(P\uparrow Q\right)\Leftrightarrow\lnot P \](88)
\[ P\nleftarrow\left(P\nrightarrow Q\right)\Leftrightarrow0 \](89)
\[ P\nleftarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\nleftarrow Q \](90)
\[ P\nleftarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](91)
\[ P\nleftrightarrow\left(P\lor Q\right)\Leftrightarrow P\nleftarrow Q \](92)
\[ P\nleftrightarrow\left(P\land Q\right)\Leftrightarrow P\nrightarrow Q \](93)
\[ P\nleftrightarrow\left(P\rightarrow Q\right)\Leftrightarrow P\uparrow Q \](94)
\[ P\nleftrightarrow\left(P\leftarrow Q\right)\Leftrightarrow P\downarrow Q \](95)
\[ P\nleftrightarrow\left(P\leftrightarrow Q\right)\Leftrightarrow\lnot Q \](96)
\[ P\nleftrightarrow\left(P\downarrow Q\right)\Leftrightarrow P\leftarrow Q \](97)
\[ P\nleftrightarrow\left(P\uparrow Q\right)\Leftrightarrow P\rightarrow Q \](98)
\[ P\nleftrightarrow\left(P\nrightarrow Q\right)\Leftrightarrow P\land Q \](99)
\[ P\nleftrightarrow\left(P\nleftarrow Q\right)\Leftrightarrow P\lor Q \](100)
\[ P\nleftrightarrow\left(P\nleftrightarrow Q\right)\Leftrightarrow Q \]肯定と否定パターン
(101)
\[ P\lor\left(\lnot P\lor Q\right)\Leftrightarrow1 \](102)
\[ P\lor\left(\lnot P\land Q\right)\Leftrightarrow P\lor Q \](103)
\[ P\lor\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\lor Q \](104)
\[ P\lor\left(\lnot P\leftarrow Q\right)\Leftrightarrow1 \](105)
\[ P\lor\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\lor Q \](106)
\[ P\lor\left(\lnot P\downarrow Q\right)\Leftrightarrow P \](107)
\[ P\lor\left(\lnot P\uparrow Q\right)\Leftrightarrow \](108)
\[ P\lor\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](109)
\[ P\lor\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P \](110)
\[ P\lor\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](111)
\[ P\land\left(\lnot P\lor Q\right)\Leftrightarrow P\land Q \](112)
\[ P\land\left(\lnot P\land Q\right)\Leftrightarrow0 \](113)
\[ P\land\left(\lnot P\rightarrow Q\right)\Leftrightarrow P \](114)
\[ P\land\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\nrightarrow Q \](115)
\[ P\land\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](116)
\[ P\land\left(\lnot P\downarrow Q\right)\Leftrightarrow P\nrightarrow Q \](117)
\[ P\land\left(\lnot P\uparrow Q\right)\Leftrightarrow P \](118)
\[ P\land\left(\lnot P\nrightarrow Q\right)\Leftrightarrow0 \](119)
\[ P\land\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\land Q \](120)
\[ P\land\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\land Q \](121)
\[ P\rightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\rightarrow Q \](122)
\[ P\rightarrow\left(\lnot P\land Q\right)\Leftrightarrow\lnot P \](123)
\[ P\rightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow1 \](124)
\[ P\rightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\uparrow Q \](125)
\[ P\rightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\uparrow Q \](126)
\[ P\rightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\uparrow Q \](127)
\[ P\rightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow1 \](128)
\[ P\rightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow\lnot P \](129)
\[ P\rightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\rightarrow Q \](130)
\[ P\rightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \](131)
\[ P\leftarrow\left(\lnot P\lor Q\right)\Leftrightarrow P \](132)
\[ P\leftarrow\left(\lnot P\land Q\right)\Leftrightarrow P\leftarrow Q \](133)
\[ P\leftarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\leftarrow Q \](134)
\[ P\leftarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P \](135)
\[ P\leftarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](136)
\[ P\leftarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow1 \](137)
\[ P\leftarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\lor Q \](138)
\[ P\leftarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\lor Q \](139)
\[ P\leftarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow1 \](140)
\[ P\leftarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\lor Q \](141)
\[ P\leftrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\land Q \](142)
\[ P\leftrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P\downarrow Q \](143)
\[ P\leftrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\leftarrow Q \](144)
\[ P\leftrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\nrightarrow Q \](145)
\[ P\leftrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow\lnot Q \](146)
\[ P\leftrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\uparrow Q \](147)
\[ P\leftrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\lor Q \](148)
\[ P\leftrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](149)
\[ P\leftrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\rightarrow Q \](150)
\[ P\leftrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow Q \](151)
\[ P\downarrow\left(\lnot P\lor Q\right)\Leftrightarrow0 \](152)
\[ P\downarrow\left(\lnot P\land Q\right)\Leftrightarrow P\downarrow Q \](153)
\[ P\downarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\downarrow Q \](154)
\[ P\downarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow0 \](155)
\[ P\downarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\downarrow Q \](156)
\[ P\downarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow\lnot P \](157)
\[ P\downarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow\lnot P\land Q \](158)
\[ P\downarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](159)
\[ P\downarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow\lnot P \](160)
\[ P\downarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](161)
\[ P\uparrow\left(\lnot P\lor Q\right)\Leftrightarrow P\uparrow Q \](162)
\[ P\uparrow\left(\lnot P\land Q\right)\Leftrightarrow1 \](163)
\[ P\uparrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow\lnot P \](164)
\[ P\uparrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\rightarrow Q \](165)
\[ P\uparrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\rightarrow Q \](166)
\[ P\uparrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\rightarrow Q \](167)
\[ P\uparrow\left(\lnot P\uparrow Q\right)\Leftrightarrow\lnot P \](168)
\[ P\uparrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow1 \](169)
\[ P\uparrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\uparrow Q \](170)
\[ P\uparrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\uparrow Q \](171)
\[ P\nrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\nrightarrow Q \](172)
\[ P\nrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P \](173)
\[ P\nrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow0 \](174)
\[ P\nrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\land Q \](175)
\[ P\nrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\land Q \](176)
\[ P\nrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\land Q \](177)
\[ P\nrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow0 \](178)
\[ P\nrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P \](179)
\[ P\nrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\nrightarrow Q \](180)
\[ P\nrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\nrightarrow Q \](181)
\[ P\nleftarrow\left(\lnot P\lor Q\right)\Leftrightarrow\lnot P \](182)
\[ P\nleftarrow\left(\lnot P\land Q\right)\Leftrightarrow P\nleftarrow Q \](183)
\[ P\nleftarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](184)
\[ P\nleftarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow\lnot P \](185)
\[ P\nleftarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](186)
\[ P\nleftarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow0 \](187)
\[ P\nleftarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\downarrow Q \](188)
\[ P\nleftarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\downarrow Q \](189)
\[ P\nleftarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow0 \](190)
\[ P\nleftarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow P\downarrow Q \](191)
\[ P\nleftrightarrow\left(\lnot P\lor Q\right)\Leftrightarrow P\uparrow Q \](192)
\[ P\nleftrightarrow\left(\lnot P\land Q\right)\Leftrightarrow P\lor Q \](193)
\[ P\nleftrightarrow\left(\lnot P\rightarrow Q\right)\Leftrightarrow P\nleftarrow Q \](194)
\[ P\nleftrightarrow\left(\lnot P\leftarrow Q\right)\Leftrightarrow P\rightarrow Q \](195)
\[ P\nleftrightarrow\left(\lnot P\leftrightarrow Q\right)\Leftrightarrow Q \](196)
\[ P\nleftrightarrow\left(\lnot P\downarrow Q\right)\Leftrightarrow P\land Q \](197)
\[ P\nleftrightarrow\left(\lnot P\uparrow Q\right)\Leftrightarrow P\downarrow Q \](198)
\[ P\nleftrightarrow\left(\lnot P\nrightarrow Q\right)\Leftrightarrow P\leftarrow Q \](199)
\[ P\nleftrightarrow\left(\lnot P\nleftarrow Q\right)\Leftrightarrow P\nrightarrow Q \](200)
\[ P\nleftrightarrow\left(\lnot P\nleftrightarrow Q\right)\Leftrightarrow\lnot Q \]一覧
\[ \begin{array}{|c|c|c|c|} \hline L_{1} & L_{2} & PL_{1}\left(PL_{2}Q\right) & PL_{1}\left(\lnot PL_{2}Q\right)\\ \hline \lor & \lor & P\lor Q & 1\\ \hline \lor & \land & P & P\lor Q\\ \hline \lor & \rightarrow & 1 & P\lor Q\\ \hline \lor & \leftarrow & P\leftarrow Q & 1\\ \hline \lor & \leftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \lor & \downarrow & P\leftarrow Q & P\\ \hline \lor & \uparrow & 1 & P\leftarrow Q\\ \hline \lor & \nrightarrow & P & P\leftarrow Q\\ \hline \lor & \nleftarrow & P\lor Q & P\\ \hline \lor & \nleftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \land & \lor & P & P\land Q\\ \hline \land & \land & P\land Q & 0\\ \hline \land & \rightarrow & P\land Q & P\\ \hline \land & \leftarrow & P & P\nrightarrow Q\\ \hline \land & \leftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \land & \downarrow & 0 & P\nrightarrow Q\\ \hline \land & \uparrow & P\nrightarrow Q & P\\ \hline \land & \nrightarrow & P\nrightarrow Q & 0\\ \hline \land & \nleftarrow & 0 & P\land Q\\ \hline \land & \nleftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \rightarrow & \lor & 1 & P\rightarrow Q\\ \hline \rightarrow & \land & P\rightarrow Q & \lnot P\\ \hline \rightarrow & \rightarrow & P\rightarrow Q & 1\\ \hline \rightarrow & \leftarrow & 1 & P\uparrow Q\\ \hline \rightarrow & \leftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \rightarrow & \downarrow & \lnot P & P\uparrow Q\\ \hline \rightarrow & \uparrow & P\uparrow Q & 1\\ \hline \rightarrow & \nrightarrow & P\uparrow Q & \lnot P\\ \hline \rightarrow & \nleftarrow & \lnot P & P\rightarrow Q\\ \hline \rightarrow & \nleftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \leftarrow & \lor & P\leftarrow Q & P\\ \hline \leftarrow & \land & 1 & P\leftarrow Q\\ \hline \leftarrow & \rightarrow & P & P\leftarrow Q\\ \hline \leftarrow & \leftarrow & P\lor Q & P\\ \hline \leftarrow & \leftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \leftarrow & \downarrow & P\lor Q & 1\\ \hline \leftarrow & \uparrow & P & P\lor Q\\ \hline \leftarrow & \nrightarrow & 1 & P\lor Q\\ \hline \leftarrow & \nleftarrow & P\leftarrow Q & 1\\ \hline \leftarrow & \nleftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \leftrightarrow & \lor & P\leftarrow Q & P\land Q\\ \hline \leftrightarrow & \land & P\rightarrow Q & P\downarrow Q\\ \hline \leftrightarrow & \rightarrow & P\land Q & P\leftarrow Q\\ \hline \leftrightarrow & \leftarrow & P\lor Q & P\nrightarrow Q\\ \hline \leftrightarrow & \leftrightarrow & Q & \lnot Q\\ \hline \leftrightarrow & \downarrow & P\nleftarrow Q & P\uparrow Q\\ \hline \leftrightarrow & \uparrow & P\nrightarrow Q & P\lor Q\\ \hline \leftrightarrow & \nrightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \leftrightarrow & \nleftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \leftrightarrow & \nleftrightarrow & \lnot Q & Q\\ \hline \downarrow & \lor & P\downarrow Q & 0\\ \hline \downarrow & \land & \lnot P & P\downarrow Q\\ \hline \downarrow & \rightarrow & 0 & P\downarrow Q\\ \hline \downarrow & \leftarrow & P\nleftarrow Q & 0\\ \hline \downarrow & \leftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \downarrow & \downarrow & P\nleftarrow Q & \lnot P\\ \hline \downarrow & \uparrow & 0 & \lnot P\land Q\\ \hline \downarrow & \nrightarrow & \lnot P & P\nleftarrow Q\\ \hline \downarrow & \nleftarrow & P\downarrow Q & \lnot P\\ \hline \downarrow & \nleftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \uparrow & \lor & \lnot P & P\uparrow Q\\ \hline \uparrow & \land & P\uparrow Q & 1\\ \hline \uparrow & \rightarrow & P\uparrow Q & \lnot P\\ \hline \uparrow & \leftarrow & \lnot P & P\rightarrow Q\\ \hline \uparrow & \leftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \uparrow & \downarrow & 1 & P\rightarrow Q\\ \hline \uparrow & \uparrow & P\rightarrow Q & \lnot P\\ \hline \uparrow & \nrightarrow & P\rightarrow Q & 1\\ \hline \uparrow & \nleftarrow & 1 & P\uparrow Q\\ \hline \uparrow & \nleftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \nrightarrow & \lor & 0 & P\nrightarrow Q\\ \hline \nrightarrow & \land & P\nrightarrow Q & P\\ \hline \nrightarrow & \rightarrow & P\nrightarrow Q & 0\\ \hline \nrightarrow & \leftarrow & 0 & P\land Q\\ \hline \nrightarrow & \leftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \nrightarrow & \downarrow & P & P\land Q\\ \hline \nrightarrow & \uparrow & P\land Q & 0\\ \hline \nrightarrow & \nrightarrow & P\land Q & P\\ \hline \nrightarrow & \nleftarrow & P & P\nrightarrow Q\\ \hline \nrightarrow & \nleftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \nleftarrow & \lor & P\nleftarrow Q & \lnot P\\ \hline \nleftarrow & \land & 0 & P\nleftarrow Q\\ \hline \nleftarrow & \rightarrow & \lnot P & P\nleftarrow Q\\ \hline \nleftarrow & \leftarrow & P\downarrow Q & \lnot P\\ \hline \nleftarrow & \leftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \nleftarrow & \downarrow & P\downarrow Q & 0\\ \hline \nleftarrow & \uparrow & \lnot P & P\downarrow Q\\ \hline \nleftarrow & \nrightarrow & 0 & P\downarrow Q\\ \hline \nleftarrow & \nleftarrow & P\nleftarrow Q & 0\\ \hline \nleftarrow & \nleftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \lor & P\nleftarrow Q & P\uparrow Q\\ \hline \nleftrightarrow & \land & P\nrightarrow Q & P\lor Q\\ \hline \nleftrightarrow & \rightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \nleftrightarrow & \leftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \nleftrightarrow & \leftrightarrow & \lnot Q & Q\\ \hline \nleftrightarrow & \downarrow & P\leftarrow Q & P\land Q\\ \hline \nleftrightarrow & \uparrow & P\rightarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \nrightarrow & P\land Q & P\leftarrow Q\\ \hline \nleftrightarrow & \nleftarrow & P\lor Q & P\nrightarrow Q\\ \hline \nleftrightarrow & \nleftrightarrow & Q & \lnot Q \\\hline \end{array} \]
\[ \begin{array}{|c|c|c|c|} \hline L_{1} & L_{2} & PL_{1}\left(PL_{2}Q\right) & PL_{1}\left(\lnot PL_{2}Q\right)\\ \hline \lor & \lor & P\lor Q & 1\\ \hline \lor & \land & P & P\lor Q\\ \hline \lor & \rightarrow & 1 & P\lor Q\\ \hline \lor & \leftarrow & P\leftarrow Q & 1\\ \hline \lor & \leftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \lor & \downarrow & P\leftarrow Q & P\\ \hline \lor & \uparrow & 1 & P\leftarrow Q\\ \hline \lor & \nrightarrow & P & P\leftarrow Q\\ \hline \lor & \nleftarrow & P\lor Q & P\\ \hline \lor & \nleftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \land & \lor & P & P\land Q\\ \hline \land & \land & P\land Q & 0\\ \hline \land & \rightarrow & P\land Q & P\\ \hline \land & \leftarrow & P & P\nrightarrow Q\\ \hline \land & \leftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \land & \downarrow & 0 & P\nrightarrow Q\\ \hline \land & \uparrow & P\nrightarrow Q & P\\ \hline \land & \nrightarrow & P\nrightarrow Q & 0\\ \hline \land & \nleftarrow & 0 & P\land Q\\ \hline \land & \nleftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \rightarrow & \lor & 1 & P\rightarrow Q\\ \hline \rightarrow & \land & P\rightarrow Q & \lnot P\\ \hline \rightarrow & \rightarrow & P\rightarrow Q & 1\\ \hline \rightarrow & \leftarrow & 1 & P\uparrow Q\\ \hline \rightarrow & \leftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \rightarrow & \downarrow & \lnot P & P\uparrow Q\\ \hline \rightarrow & \uparrow & P\uparrow Q & 1\\ \hline \rightarrow & \nrightarrow & P\uparrow Q & \lnot P\\ \hline \rightarrow & \nleftarrow & \lnot P & P\rightarrow Q\\ \hline \rightarrow & \nleftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \leftarrow & \lor & P\leftarrow Q & P\\ \hline \leftarrow & \land & 1 & P\leftarrow Q\\ \hline \leftarrow & \rightarrow & P & P\leftarrow Q\\ \hline \leftarrow & \leftarrow & P\lor Q & P\\ \hline \leftarrow & \leftrightarrow & P\lor Q & P\leftarrow Q\\ \hline \leftarrow & \downarrow & P\lor Q & 1\\ \hline \leftarrow & \uparrow & P & P\lor Q\\ \hline \leftarrow & \nrightarrow & 1 & P\lor Q\\ \hline \leftarrow & \nleftarrow & P\leftarrow Q & 1\\ \hline \leftarrow & \nleftrightarrow & P\leftarrow Q & P\lor Q\\ \hline \leftrightarrow & \lor & P\leftarrow Q & P\land Q\\ \hline \leftrightarrow & \land & P\rightarrow Q & P\downarrow Q\\ \hline \leftrightarrow & \rightarrow & P\land Q & P\leftarrow Q\\ \hline \leftrightarrow & \leftarrow & P\lor Q & P\nrightarrow Q\\ \hline \leftrightarrow & \leftrightarrow & Q & \lnot Q\\ \hline \leftrightarrow & \downarrow & P\nleftarrow Q & P\uparrow Q\\ \hline \leftrightarrow & \uparrow & P\nrightarrow Q & P\lor Q\\ \hline \leftrightarrow & \nrightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \leftrightarrow & \nleftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \leftrightarrow & \nleftrightarrow & \lnot Q & Q\\ \hline \downarrow & \lor & P\downarrow Q & 0\\ \hline \downarrow & \land & \lnot P & P\downarrow Q\\ \hline \downarrow & \rightarrow & 0 & P\downarrow Q\\ \hline \downarrow & \leftarrow & P\nleftarrow Q & 0\\ \hline \downarrow & \leftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \downarrow & \downarrow & P\nleftarrow Q & \lnot P\\ \hline \downarrow & \uparrow & 0 & \lnot P\land Q\\ \hline \downarrow & \nrightarrow & \lnot P & P\nleftarrow Q\\ \hline \downarrow & \nleftarrow & P\downarrow Q & \lnot P\\ \hline \downarrow & \nleftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \uparrow & \lor & \lnot P & P\uparrow Q\\ \hline \uparrow & \land & P\uparrow Q & 1\\ \hline \uparrow & \rightarrow & P\uparrow Q & \lnot P\\ \hline \uparrow & \leftarrow & \lnot P & P\rightarrow Q\\ \hline \uparrow & \leftrightarrow & P\uparrow Q & P\rightarrow Q\\ \hline \uparrow & \downarrow & 1 & P\rightarrow Q\\ \hline \uparrow & \uparrow & P\rightarrow Q & \lnot P\\ \hline \uparrow & \nrightarrow & P\rightarrow Q & 1\\ \hline \uparrow & \nleftarrow & 1 & P\uparrow Q\\ \hline \uparrow & \nleftrightarrow & P\rightarrow Q & P\uparrow Q\\ \hline \nrightarrow & \lor & 0 & P\nrightarrow Q\\ \hline \nrightarrow & \land & P\nrightarrow Q & P\\ \hline \nrightarrow & \rightarrow & P\nrightarrow Q & 0\\ \hline \nrightarrow & \leftarrow & 0 & P\land Q\\ \hline \nrightarrow & \leftrightarrow & P\nrightarrow Q & P\land Q\\ \hline \nrightarrow & \downarrow & P & P\land Q\\ \hline \nrightarrow & \uparrow & P\land Q & 0\\ \hline \nrightarrow & \nrightarrow & P\land Q & P\\ \hline \nrightarrow & \nleftarrow & P & P\nrightarrow Q\\ \hline \nrightarrow & \nleftrightarrow & P\land Q & P\nrightarrow Q\\ \hline \nleftarrow & \lor & P\nleftarrow Q & \lnot P\\ \hline \nleftarrow & \land & 0 & P\nleftarrow Q\\ \hline \nleftarrow & \rightarrow & \lnot P & P\nleftarrow Q\\ \hline \nleftarrow & \leftarrow & P\downarrow Q & \lnot P\\ \hline \nleftarrow & \leftrightarrow & P\downarrow Q & P\nleftarrow Q\\ \hline \nleftarrow & \downarrow & P\downarrow Q & 0\\ \hline \nleftarrow & \uparrow & \lnot P & P\downarrow Q\\ \hline \nleftarrow & \nrightarrow & 0 & P\downarrow Q\\ \hline \nleftarrow & \nleftarrow & P\nleftarrow Q & 0\\ \hline \nleftarrow & \nleftrightarrow & P\nleftarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \lor & P\nleftarrow Q & P\uparrow Q\\ \hline \nleftrightarrow & \land & P\nrightarrow Q & P\lor Q\\ \hline \nleftrightarrow & \rightarrow & P\uparrow Q & P\nleftarrow Q\\ \hline \nleftrightarrow & \leftarrow & P\downarrow Q & P\rightarrow Q\\ \hline \nleftrightarrow & \leftrightarrow & \lnot Q & Q\\ \hline \nleftrightarrow & \downarrow & P\leftarrow Q & P\land Q\\ \hline \nleftrightarrow & \uparrow & P\rightarrow Q & P\downarrow Q\\ \hline \nleftrightarrow & \nrightarrow & P\land Q & P\leftarrow Q\\ \hline \nleftrightarrow & \nleftarrow & P\lor Q & P\nrightarrow Q\\ \hline \nleftrightarrow & \nleftrightarrow & Q & \lnot Q \\\hline \end{array} \]
(1)
\begin{align*} P\lor\left(P\lor Q\right) & \Leftrightarrow\left(P\lor P\right)\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(2)
吸収律より、\[ P\lor\left(P\land Q\right)\Leftrightarrow P \] が成り立つ。
(3)
\begin{align*} P\lor\left(P\rightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow1\lor Q\\ & \Leftrightarrow1 \end{align*}(4)
\begin{align*} P\lor\left(P\leftarrow Q\right) & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(5)
\begin{align*} P\lor\left(P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(6)
\begin{align*} P\lor\left(P\downarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(7)
\begin{align*} P\lor\left(P\uparrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(8)
\begin{align*} P\lor\left(P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(9)
\begin{align*} P\lor\left(P\nleftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(10)
\begin{align*} P\lor\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(11)
吸収律より、\[ P\land\left(P\lor Q\right)\Leftrightarrow P \] が成り立つ。
(12)
\begin{align*} P\land\left(P\land Q\right) & \Leftrightarrow\left(P\land P\right)\land Q\\ & \Leftrightarrow P\land Q \end{align*}(13)
\begin{align*} P\land\left(P\rightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(14)
\begin{align*} P\land\left(P\leftarrow Q\right) & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(15)
\begin{align*} P\land\left(P\leftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(16)
\begin{align*} P\land\left(P\downarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(17)
\begin{align*} P\land\left(P\uparrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(18)
\begin{align*} P\land\left(P\nrightarrow Q\right) & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(19)
\begin{align*} P\land\left(P\nleftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land Q\\ & \Leftrightarrow0 \end{align*}(20)
\begin{align*} P\land\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(21)
\begin{align*} P\rightarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}(22)
\begin{align*} P\rightarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(23)
\begin{align*} P\rightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(24)
\begin{align*} P\rightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(25)
\begin{align*} P\rightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(26)
\begin{align*} P\rightarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(27)
\begin{align*} P\rightarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(28)
\begin{align*} P\rightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(29)
\begin{align*} P\rightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(30)
\begin{align*} P\rightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(31)
\begin{align*} P\leftarrow\left(P\lor Q\right) & \Leftrightarrow P\lor\lnot\left(P\lor Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(32)
\begin{align*} P\leftarrow\left(P\land Q\right) & \Leftrightarrow P\lor\lnot\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(33)
\begin{align*} P\leftarrow\left(P\rightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(34)
\begin{align*} P\leftarrow\left(P\leftarrow Q\right) & \Leftrightarrow P\lor\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(35)
\begin{align*} P\leftarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(36)
\begin{align*} P\leftarrow\left(P\downarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(37)
\begin{align*} P\leftarrow\left(P\uparrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\uparrow Q\right)\\ & \Leftrightarrow P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}(38)
\begin{align*} P\leftarrow\left(P\nrightarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\nrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\rightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}(39)
\begin{align*} P\leftarrow\left(P\nleftarrow Q\right) & \Leftrightarrow P\lor\lnot\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\leftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(40)
\begin{align*} P\leftarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(41)
\begin{align*} P\leftrightarrow\left(P\lor Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\lor Q\right)\right\} \land\left\{ P\lor\lnot\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor P\right)\lor Q\right\} \land\left\{ P\lor\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(42)
\begin{align*} P\leftrightarrow\left(P\land Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\land Q\right)\right\} \land\left\{ P\lor\lnot\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ P\lor\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ \left(P\lor\lnot P\right)\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(43)
\begin{align*} P\leftrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\rightarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\lor Q\right)\right\} \land\left\{ P\lor\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor\lnot P\right)\lor Q\right\} \land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land P\\ & \Leftrightarrow P\land Q \end{align*}(44)
\begin{align*} P\leftrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\leftarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\lor\lnot Q\right)\right\} \land\left\{ P\lor\left(P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor P\right)\lor\lnot Q\right\} \land\left\{ P\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(45)
\begin{align*} P\leftrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\left(P\leftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow1\leftrightarrow Q\\ & \Leftrightarrow Q \end{align*}(46)
\begin{align*} P\leftrightarrow\left(P\downarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\downarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land\lnot Q\right)\right\} \land\left\{ P\lor\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor P\right)\lor Q\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(47)
\begin{align*} P\leftrightarrow\left(P\uparrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\uparrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\lor\lnot Q\right)\right\} \land\left\{ P\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(\lnot P\lor\lnot P\right)\lor\lnot Q\right\} \land P\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land P\\ & \Leftrightarrow\lnot Q\land P\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(48)
\begin{align*} P\leftrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\nrightarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land\lnot Q\right)\right\} \land\left\{ P\lor\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left\{ P\lor\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left\{ \left(P\lor\lnot P\right)\lor Q\right\} \\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\uparrow Q \end{align*}(49)
\begin{align*} P\leftrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(P\nleftarrow Q\right)\right\} \land\left\{ P\lor\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land Q\right)\right\} \land\left\{ P\lor\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor P\right)\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(50)
\begin{align*} P\leftrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\leftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow1\nleftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}(51)
\begin{align*} P\downarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(52)
\begin{align*} P\downarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(53)
\begin{align*} P\downarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(54)
\begin{align*} P\downarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(55)
\begin{align*} P\downarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(56)
\begin{align*} P\downarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(57)
\begin{align*} P\downarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land Q\\ & \Leftrightarrow0 \end{align*}(58)
\begin{align*} P\downarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(59)
\begin{align*} P\downarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(60)
\begin{align*} P\downarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(61)
\begin{align*} P\uparrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(62)
\begin{align*} P\uparrow\left(P\land Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(63)
\begin{align*} P\uparrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(64)
\begin{align*} P\uparrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(65)
\begin{align*} P\uparrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(66)
\begin{align*} P\uparrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor Q\\ & \Leftrightarrow1 \end{align*}(67)
\begin{align*} P\uparrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(68)
\begin{align*} P\uparrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(69)
\begin{align*} P\uparrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\lor P\right)\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(70)
\begin{align*} P\uparrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(\lnot P\lor Q\right)\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot P\right)\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(71)
\begin{align*} P\nrightarrow\left(P\lor Q\right) & \Leftrightarrow P\land\lnot\left(P\lor Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(72)
\begin{align*} P\nrightarrow\left(P\land Q\right) & \Leftrightarrow P\land\lnot\left(P\land Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(73)
\begin{align*} P\nrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow P\land\left(P\nrightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(74)
\begin{align*} P\nrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow P\land\left(P\nleftarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(P\land\lnot P\right)\land Q\\ & \Leftrightarrow0 \end{align*}(75)
\begin{align*} P\nrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(76)
\begin{align*} P\nrightarrow\left(P\downarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}(77)
\begin{align*} P\nrightarrow\left(P\uparrow Q\right) & \Leftrightarrow P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(P\land P\right)\land Q\\ & \Leftrightarrow P\land Q \end{align*}(78)
\begin{align*} P\nrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow P\land\left(P\rightarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(79)
\begin{align*} P\nrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow P\land\left(P\leftarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(80)
\begin{align*} P\nrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left(P\leftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(\lnot P\lor Q\right)\lor\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(81)
\begin{align*} P\nleftarrow\left(P\lor Q\right) & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(82)
\begin{align*} P\nleftarrow\left(P\land Q\right) & \Leftrightarrow\lnot P\land\left(P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land Q\\ & \Leftrightarrow0 \end{align*}(83)
\begin{align*} P\nleftarrow\left(P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(84)
\begin{align*} P\nleftarrow\left(P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(85)
\begin{align*} P\nleftarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left\{ \left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(86)
\begin{align*} P\nleftarrow\left(P\downarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(87)
\begin{align*} P\nleftarrow\left(P\uparrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(88)
\begin{align*} P\nleftarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(P\land\lnot Q\right)\\ & \Leftrightarrow\left(\lnot P\land P\right)\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(89)
\begin{align*} P\nleftarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(90)
\begin{align*} P\nleftarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left\{ \left(P\land\lnot Q\right)\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot P\right)\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(91)
\begin{align*} P\nleftrightarrow\left(P\lor Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\land\lnot Q\right)\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\left\{ \left(P\land\lnot P\right)\land\lnot Q\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(92)
\begin{align*} P\nleftrightarrow\left(P\land Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor\lnot Q\right)\right\} \lor\left\{ \left(\lnot P\land P\right)\land Q\right\} \\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(93)
\begin{align*} P\nleftrightarrow\left(P\rightarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\rightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\land\lnot Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land P\right)\land\lnot Q\right\} \lor\lnot P\\ & \Leftrightarrow\left\{ P\land\lnot Q\right\} \lor\lnot P\\ & \Leftrightarrow\lnot Q\lor\lnot P\\ & \Leftrightarrow P\uparrow Q \end{align*}(94)
\begin{align*} P\nleftrightarrow\left(P\leftarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\leftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land\lnot P\right)\land Q\right\} \lor\left\{ \lnot P\land\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(95)
\begin{align*} P\nleftrightarrow\left(P\leftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow0\leftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}(96)
\begin{align*} P\nleftrightarrow\left(P\downarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\downarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\land\lnot P\right)\land\lnot Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(97)
\begin{align*} P\nleftrightarrow\left(P\uparrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\uparrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\land Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \left(P\land P\right)\land Q\right\} \lor\lnot P\\ & \Leftrightarrow\left(P\land Q\right)\lor\lnot P\\ & \Leftrightarrow Q\lor\lnot P\\ & \Leftrightarrow P\rightarrow Q \end{align*}(98)
\begin{align*} P\nleftrightarrow\left(P\nrightarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\nrightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land Q\right\} \lor\left\{ \left(\lnot P\land P\right)\land\lnot Q\right\} \\ & \Leftrightarrow P\land Q \end{align*}(99)
\begin{align*} P\nleftrightarrow\left(P\nleftarrow Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(P\nleftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor\lnot Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left\{ \left(\lnot P\land\lnot P\right)\land Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(100)
\begin{align*} P\nleftrightarrow\left(P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow0\nleftrightarrow Q\\ & \Leftrightarrow Q \end{align*}(101)
\begin{align*} P\lor\left(\lnot P\lor Q\right) & \Leftrightarrow P\lor\lnot P\lor Q\\ & \Leftrightarrow1 \end{align*}(102)
\begin{align*} P\lor\left(\lnot P\land Q\right) & \Leftrightarrow\left(P\lor\lnot P\right)\land\left(P\lor Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(103)
\begin{align*} P\lor\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\lor\left(P\lor Q\right)\\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(104)
\begin{align*} P\lor\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(105)
\begin{align*} P\lor\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(106)
\begin{align*} P\lor\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(107)
\begin{align*} P\lor\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\lor P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(108)
\begin{align*} P\lor\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(109)
\begin{align*} P\lor\left(\lnot P\nleftarrow Q\right) & P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}(110)
\begin{align*} P\lor\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(111)
\begin{align*} P\land\left(\lnot P\lor Q\right) & \Leftrightarrow\left(P\land\lnot P\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(112)
\begin{align*} P\land\left(\lnot P\land Q\right) & \Leftrightarrow P\land\lnot P\land Q\\ & \Leftrightarrow0 \end{align*}(113)
\begin{align*} P\land\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}(114)
\begin{align*} P\land\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(115)
\begin{align*} P\land\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(116)
\begin{align*} P\land\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\land P\land\lnot Q\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(117)
\begin{align*} P\land\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(118)
\begin{align*} P\land\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\land\lnot P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(119)
\begin{align*} P\land\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}(120)
\begin{align*} P\land\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}(121)
\begin{align*} P\rightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot P\lor Q\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(122)
\begin{align*} P\rightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(123)
\begin{align*} P\rightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor P\lor Q\\ & \Leftrightarrow1 \end{align*}(124)
\begin{align*} P\rightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(125)
\begin{align*} P\rightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(126)
\begin{align*} P\rightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(127)
\begin{align*} P\rightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\lor P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(128)
\begin{align*} P\rightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(129)
\begin{align*} P\rightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(130)
\begin{align*} P\rightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(131)
\begin{align*} P\leftarrow\left(\lnot P\lor Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(132)
\begin{align*} P\leftarrow\left(\lnot P\land Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor P\lor\lnot Q\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(133)
\begin{align*} P\leftarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(134)
\begin{align*} P\leftarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow P\lor\left(P\land Q\right)\\ & \Leftrightarrow P \end{align*}(135)
\begin{align*} P\leftarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(136)
\begin{align*} P\leftarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow P\lor\lnot P\lor Q\\ & \Leftrightarrow1 \end{align*}(137)
\begin{align*} P\leftarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\lor\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(138)
\begin{align*} P\leftarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(139)
\begin{align*} P\leftarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(140)
\begin{align*} P\leftarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\lor\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor P\lor Q\\ & \Leftrightarrow P\lor Q \end{align*}(141)
\begin{align*} P\leftrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\left\{ \lnot P\lor\lnot P\lor Q\right\} \land\left\{ P\lor\lnot\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor Q\right\} \land\left\{ P\lor\left(P\land\lnot Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor Q\right\} \land P\\ & \Leftrightarrow P\land Q \end{align*}(142)
\begin{align*} P\leftrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(143)
\begin{align*} P\leftrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\rightarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor P\lor Q\right\} \land\left\{ P\lor\left(\lnot P\land\lnot Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(144)
\begin{align*} P\leftrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\leftarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\lnot P\lor\lnot Q\right\} \land\left\{ P\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\lnot Q\right\} \land P\\ & \Leftrightarrow\lnot Q\land P\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(145)
\begin{align*} P\leftrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\leftrightarrow\lnot P\leftrightarrow Q\\ & \Leftrightarrow0\leftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}(146)
\begin{align*} P\leftrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\downarrow Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land\lnot Q\right)\right\} \land\left\{ P\lor\lnot P\lor Q\right\} \\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(147)
\begin{align*} P\leftrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\uparrow Q\right)\right\} \land\left\{ P\lor\lnot\left(\lnot P\uparrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor P\lor\lnot Q\right\} \land\left\{ P\lor\left(\lnot P\land Q\right)\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(148)
\begin{align*} P\leftrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\nrightarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\land\lnot Q\right)\right\} \land\left\{ P\lor P\lor Q\right\} \\ & \Leftrightarrow\lnot P\land\left\{ P\lor Q\right\} \\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(149)
\begin{align*} P\leftrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\left\{ \lnot P\lor\left(\lnot P\nleftarrow Q\right)\right\} \land\left\{ P\lor\left(\lnot P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ \lnot P\lor\left(P\land Q\right)\right\} \land\left\{ P\lor\lnot P\lor\lnot Q\right\} \\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(150)
\begin{align*} P\leftrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\leftrightarrow\lnot P\nleftrightarrow Q\\ & \Leftrightarrow0\nleftrightarrow Q\\ & \Leftrightarrow Q \end{align*}(151)
\begin{align*} P\downarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\land P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(152)
\begin{align*} P\downarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(153)
\begin{align*} P\downarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(154)
\begin{align*} P\downarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land Q\\ & \Leftrightarrow0 \end{align*}(155)
\begin{align*} P\downarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(156)
\begin{align*} P\downarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(157)
\begin{align*} P\downarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\land\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land Q\\ & \Leftrightarrow\lnot P\land Q \end{align*}(158)
\begin{align*} P\downarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(159)
\begin{align*} P\downarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(160)
\begin{align*} P\downarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(161)
\begin{align*} P\uparrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(162)
\begin{align*} P\uparrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P\lor P\lor\lnot Q\\ & \Leftrightarrow1 \end{align*}(163)
\begin{align*} P\uparrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(164)
\begin{align*} P\uparrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(165)
\begin{align*} P\uparrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor\left(P\land Q\right)\\ & \Leftrightarrow\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(166)
\begin{align*} P\uparrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot P\lor Q\\ & \Leftrightarrow P\rightarrow Q \end{align*}(167)
\begin{align*} P\uparrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\lor\lnot\left(\lnot P\uparrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(168)
\begin{align*} P\uparrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor P\lor Q\\ & \Leftrightarrow1 \end{align*}(169)
\begin{align*} P\uparrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(170)
\begin{align*} P\uparrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\lor\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\lor\left\{ \left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\lnot P\lor\lnot P\lor\lnot Q\\ & \Leftrightarrow P\uparrow Q \end{align*}(171)
\begin{align*} P\nrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow P\land\lnot\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(172)
\begin{align*} P\nrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow P\land\lnot\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow P \end{align*}(173)
\begin{align*} P\nrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow P\land\lnot P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(174)
\begin{align*} P\nrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}(175)
\begin{align*} P\nrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow P\land P\land Q\\ & \Leftrightarrow P\land Q \end{align*}(176)
\begin{align*} P\nrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(177)
\begin{align*} P\nrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow P\land\lnot P\land Q\\ & \Leftrightarrow0 \end{align*}(178)
\begin{align*} P\nrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor Q\right)\\ & \Leftrightarrow P \end{align*}(179)
\begin{align*} P\nrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(180)
\begin{align*} P\nrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(181)
\begin{align*} P\nleftarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\lor Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(182)
\begin{align*} P\nleftarrow\left(\lnot P\land Q\right) & \Leftrightarrow\lnot P\land\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(183)
\begin{align*} P\nleftarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\rightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(184)
\begin{align*} P\nleftarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P \end{align*}(185)
\begin{align*} P\nleftarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\leftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(186)
\begin{align*} P\nleftarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\downarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land\lnot Q\\ & \Leftrightarrow0 \end{align*}(187)
\begin{align*} P\nleftarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(188)
\begin{align*} P\nleftarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(189)
\begin{align*} P\nleftarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftarrow Q\right)\\ & \Leftrightarrow\lnot P\land P\land Q\\ & \Leftrightarrow0 \end{align*}(190)
\begin{align*} P\nleftarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\lnot P\land\left(\lnot P\nleftrightarrow Q\right)\\ & \Leftrightarrow\lnot P\land\left\{ \left(\lnot P\land\lnot Q\right)\lor\left(P\land Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(191)
\begin{align*} P\nleftrightarrow\left(\lnot P\lor Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\lor Q\right)\right\} \\ & \Leftrightarrow\left(P\land P\land\lnot Q\right)\lor\lnot P\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\lnot P\\ & \Leftrightarrow\lnot Q\lor\lnot P\\ & \Leftrightarrow P\uparrow Q \end{align*}(192)
\begin{align*} P\nleftrightarrow\left(\lnot P\land Q\right) & \Leftrightarrow\left\{ P\land\lnot\left(\lnot P\land Q\right)\right\} \lor\left\{ \lnot P\land\lnot P\land Q\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor\lnot Q\right)\right\} \lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor\left(\lnot P\land Q\right)\\ & \Leftrightarrow P\lor Q \end{align*}(193)
\begin{align*} P\nleftrightarrow\left(\lnot P\rightarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\nrightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\rightarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land\lnot P\land\lnot Q\right)\lor\left\{ \lnot P\land\left(P\lor Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor Q\right)\\ & \Leftrightarrow\lnot P\land Q\\ & \Leftrightarrow P\nleftarrow Q \end{align*}(194)
\begin{align*} P\nleftrightarrow\left(\lnot P\leftarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\nleftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\leftarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land P\land Q\right)\lor\left\{ \lnot P\land\left(\lnot P\lor\lnot Q\right)\right\} \\ & \Leftrightarrow\left(P\land Q\right)\lor\lnot P\\ & \Leftrightarrow Q\lor\lnot P\\ & \Leftrightarrow P\rightarrow Q \end{align*}(195)
\begin{align*} P\nleftrightarrow\left(\lnot P\leftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow\lnot P\right)\leftrightarrow Q\\ & \Leftrightarrow\left(P\leftrightarrow P\right)\leftrightarrow Q\\ & \Leftrightarrow1\leftrightarrow Q\\ & \Leftrightarrow Q \end{align*}(196)
\begin{align*} P\nleftrightarrow\left(\lnot P\downarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\lor Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\downarrow Q\right)\right\} \\ & \Leftrightarrow\left(P\land Q\right)\lor\left(\lnot P\land P\land\lnot Q\right)\\ & \Leftrightarrow P\land Q \end{align*}(197)
\begin{align*} P\nleftrightarrow\left(\lnot P\uparrow Q\right) & \Leftrightarrow\left\{ P\land\lnot P\land Q\right\} \lor\left\{ \lnot P\land\left(\lnot P\uparrow Q\right)\right\} \\ & \Leftrightarrow\lnot P\land\left(P\lor\lnot Q\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\\ & \Leftrightarrow P\downarrow Q \end{align*}(198)
\begin{align*} P\nleftrightarrow\left(\lnot P\nrightarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\rightarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\nrightarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(P\lor Q\right)\right\} \lor\left\{ \lnot P\land\lnot P\land\lnot Q\right\} \\ & \Leftrightarrow P\lor\left(\lnot P\land\lnot Q\right)\\ & \Leftrightarrow P\lor\lnot Q\\ & \Leftrightarrow P\leftarrow Q \end{align*}(199)
\begin{align*} P\nleftrightarrow\left(\lnot P\nleftarrow Q\right) & \Leftrightarrow\left\{ P\land\left(\lnot P\leftarrow Q\right)\right\} \lor\left\{ \lnot P\land\left(\lnot P\nleftarrow Q\right)\right\} \\ & \Leftrightarrow\left\{ P\land\left(\lnot P\lor\lnot Q\right)\right\} \lor\left(\lnot P\land P\land Q\right)\\ & \Leftrightarrow P\land\left(\lnot P\lor\lnot Q\right)\\ & \Leftrightarrow P\land\lnot Q\\ & \Leftrightarrow P\nrightarrow Q \end{align*}(200)
\begin{align*} P\nleftrightarrow\left(\lnot P\nleftrightarrow Q\right) & \Leftrightarrow\left(P\nleftrightarrow\lnot P\right)\nleftrightarrow Q\\ & \Leftrightarrow\left(P\leftrightarrow P\right)\nleftrightarrow Q\\ & \Leftrightarrow1\nleftrightarrow Q\\ & \Leftrightarrow\lnot Q \end{align*}
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| タイトル | 2引数が同じ3引数の論理演算子 |
| URL | https://www.nomuramath.com/x90hzr1z/ |
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存在命題(論理和)と全称命題(論理積)の順序変更
\[
\exists x\in X,\forall y\in Y,P\left(x,y\right)\Rightarrow\forall y\in Y,\exists x\in X,P\left(x,y\right)
\]
3引数論理演算を別表記
\[
P\lor\left(Q\lor R\right)\Leftrightarrow P\leftarrow\left(Q\downarrow R\right)
\]
論理演算の基本
\[
P\lor\left(P\land Q\right)\Leftrightarrow P
\]
逆・裏・対偶の定義と対偶の法則
\[
\left(P\rightarrow Q\right)\Leftrightarrow\left(\lnot P\leftarrow\lnot Q\right)
\]