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
(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} \]
(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*}
ページ情報
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/ |
SNSボタン |