3つのうち1つを消したものとの包含関係

by nomura · 2022年12月15日

3つのうち1つを消したものとの包含関係
\(P,Q,R\)は命題変数とする。

(1)

\[ P\lor\left(Q\lor R\right)\Leftarrow P\lor Q \]

(2)重要

\[ P\lor\left(Q\land R\right)\Rightarrow P\lor Q \]

(3)

\[ P\lor\left(Q\rightarrow R\right)\Leftarrow P\leftarrow Q \]

(4)

\[ P\lor\left(Q\leftarrow R\right)\Leftarrow P\lor Q \]

(5)

\[ P\lor\left(Q\downarrow R\right)\Rightarrow P\leftarrow Q \]

(6)

\[ P\lor\left(Q\uparrow R\right)\Leftarrow P\leftarrow Q \]

(7)

\[ P\lor\left(Q\nrightarrow R\right)\Rightarrow P\lor Q \]

(8)

\[ P\lor\left(Q\nleftarrow R\right)\Rightarrow P\leftarrow Q \]

(9)重要

\[ P\land\left(Q\lor R\right)\Leftarrow P\land Q \]

(10)

\[ P\land\left(Q\land R\right)\Rightarrow P\land Q \]

(11)

\[ P\land\left(Q\rightarrow R\right)\Leftarrow P\nrightarrow Q \]

(12)

\[ P\land\left(Q\leftarrow R\right)\Leftarrow P\land Q \]

(13)

\[ P\land\left(Q\downarrow R\right)\Rightarrow P\nrightarrow Q \]

(14)

\[ P\land\left(Q\uparrow R\right)\Leftarrow P\nrightarrow Q \]

(15)

\[ P\land\left(Q\nrightarrow R\right)\Rightarrow P\land Q \]

(16)

\[ P\land\left(Q\nleftarrow R\right)\Rightarrow P\nrightarrow Q \]

(17)

\[ P\rightarrow\left(Q\lor R\right)\Leftarrow P\rightarrow Q \]

(18)

\[ P\rightarrow\left(Q\land R\right)\Rightarrow P\rightarrow Q \]

(19)

\[ P\rightarrow\left(Q\rightarrow R\right)\Leftarrow P\uparrow Q \]

(20)

\[ P\rightarrow\left(Q\leftarrow R\right)\Leftarrow P\rightarrow Q \]

(21)

\[ P\rightarrow\left(Q\downarrow R\right)\Rightarrow P\uparrow Q \]

(22)

\[ P\rightarrow\left(Q\uparrow R\right)\Leftarrow P\uparrow Q \]

(23)

\[ P\rightarrow\left(Q\nrightarrow R\right)\Rightarrow P\rightarrow Q \]

(24)

\[ P\rightarrow\left(Q\nleftarrow R\right)\Rightarrow P\uparrow Q \]

(25)

\[ P\leftarrow\left(Q\lor R\right)\Rightarrow P\leftarrow Q \]

(26)

\[ P\leftarrow\left(Q\land R\right)\Leftarrow P\leftarrow Q \]

(27)

\[ P\leftarrow\left(Q\rightarrow R\right)\Rightarrow P\lor Q \]

(28)

\[ P\leftarrow\left(Q\leftarrow R\right)\Rightarrow P\leftarrow Q \]

(29)

\[ P\leftarrow\left(Q\downarrow R\right)\Leftarrow P\lor Q \]

(30)

\[ P\leftarrow\left(Q\uparrow R\right)\Rightarrow P\lor Q \]

(31)

\[ P\leftarrow\left(Q\nrightarrow R\right)\Leftarrow P\leftarrow Q \]

(32)

\[ P\leftarrow\left(Q\nleftarrow R\right)\Leftarrow P\lor Q \]

(33)

\[ P\downarrow\left(Q\lor R\right)\Rightarrow P\downarrow Q \]

(34)

\[ P\downarrow\left(Q\land R\right)\Leftarrow P\downarrow Q \]

(35)

\[ P\downarrow\left(Q\rightarrow R\right)\Rightarrow P\nleftarrow Q \]

(36)

\[ P\downarrow\left(Q\leftarrow R\right)\Rightarrow P\downarrow Q \]

(37)

\[ P\downarrow\left(Q\downarrow R\right)\Leftarrow P\nleftarrow Q \]

(38)

\[ P\downarrow\left(Q\uparrow R\right)\Rightarrow P\nleftarrow Q \]

(39)

\[ P\downarrow\left(Q\nrightarrow R\right)\Leftarrow P\downarrow Q \]

(40)

\[ P\downarrow\left(Q\nleftarrow R\right)\Leftarrow P\nleftarrow Q \]

(41)

\[ P\uparrow\left(Q\lor R\right)\Rightarrow P\uparrow Q \]

(42)

\[ P\uparrow\left(Q\land R\right)\Leftarrow P\uparrow Q \]

(43)

\[ P\uparrow\left(Q\rightarrow R\right)\Rightarrow P\rightarrow Q \]

(44)

\[ P\uparrow\left(Q\leftarrow R\right)\Rightarrow P\uparrow Q \]

(45)

\[ P\uparrow\left(Q\downarrow R\right)\Leftarrow P\rightarrow Q \]

(46)

\[ P\uparrow\left(Q\uparrow R\right)\Rightarrow P\rightarrow Q \]

(47)

\[ P\uparrow\left(Q\nrightarrow R\right)\Leftarrow P\uparrow Q \]

(48)

\[ P\uparrow\left(Q\nleftarrow R\right)\Leftarrow P\rightarrow Q \]

(49)

\[ P\nrightarrow\left(Q\lor R\right)\Rightarrow P\nrightarrow Q \]

(50)

\[ P\nrightarrow\left(Q\land R\right)\Leftarrow P\nrightarrow Q \]

(51)

\[ P\nrightarrow\left(Q\rightarrow R\right)\Rightarrow P\land Q \]

(52)

\[ P\nrightarrow\left(Q\leftarrow R\right)\Rightarrow P\nrightarrow Q \]

(53)

\[ P\nrightarrow\left(Q\downarrow R\right)\Leftarrow P\land Q \]

(54)

\[ P\nrightarrow\left(Q\uparrow R\right)\Rightarrow P\land Q \]

(55)

\[ P\nrightarrow\left(Q\nrightarrow R\right)\Leftarrow P\nrightarrow Q \]

(56)

\[ P\nrightarrow\left(Q\nleftarrow R\right)\Leftarrow P\land Q \]

(57)

\[ P\nleftarrow\left(Q\lor R\right)\Leftarrow P\nleftarrow Q \]

(58)

\[ P\nleftarrow\left(Q\land R\right)\Rightarrow P\nleftarrow Q \]

(59)

\[ P\nleftarrow\left(Q\rightarrow R\right)\Leftarrow P\downarrow Q \]

(60)

\[ P\nleftarrow\left(Q\leftarrow R\right)\Leftarrow P\nleftarrow Q \]

(61)

\[ P\nleftarrow\left(Q\downarrow R\right)\Rightarrow P\downarrow Q \]

(62)

\[ P\nleftarrow\left(Q\uparrow R\right)\Leftarrow P\downarrow Q \]

(63)

\[ P\nleftarrow\left(Q\nrightarrow R\right)\Rightarrow P\nleftarrow Q \]

(64)

\[ P\nleftarrow\left(Q\nleftarrow R\right)\Rightarrow P\downarrow Q \]

(1)

\begin{align*} P\lor\left(Q\lor R\right) & \Leftrightarrow P\lor Q\lor R\\ & \Leftarrow P\lor Q \end{align*}

(2)

\begin{align*} P\lor\left(Q\land R\right) & \Leftrightarrow\left(P\lor Q\right)\land\left(P\lor R\right)\\ & \Rightarrow P\lor Q \end{align*}

(3)

\begin{align*} P\lor\left(Q\rightarrow R\right) & \Leftrightarrow P\lor\lnot Q\lor R\\ & \Leftarrow P\leftarrow Q \end{align*}

(4)

\begin{align*} P\lor\left(Q\leftarrow R\right) & \Leftrightarrow P\lor Q\lor\lnot R\\ & \Leftarrow P\lor Q \end{align*}

(5)

\begin{align*} P\lor\left(Q\downarrow R\right) & \Leftrightarrow P\lor\left(\lnot Q\land\lnot R\right)\\ & \Leftrightarrow\left(P\lor\lnot Q\right)\land\left(P\lor\lnot R\right)\\ & \Rightarrow P\leftarrow Q \end{align*}

(6)

\begin{align*} P\lor\left(Q\uparrow R\right) & \Leftrightarrow P\lor\lnot Q\lor\lnot R\\ & \Leftarrow P\leftarrow Q \end{align*}

(7)

\begin{align*} P\lor\left(Q\nrightarrow R\right) & \Leftrightarrow P\lor\left(Q\land\lnot R\right)\\ & \Leftrightarrow\left(P\lor Q\right)\land\left(P\lor\lnot R\right)\\ & \Rightarrow P\lor Q \end{align*}

(8)

\begin{align*} P\lor\left(Q\nleftarrow R\right) & \Leftrightarrow P\lor\left(\lnot Q\land R\right)\\ & \Leftrightarrow\left(P\lor\lnot Q\right)\land\left(P\lor R\right)\\ & \Rightarrow P\leftarrow Q \end{align*}

(9)

\begin{align*} P\land\left(Q\lor R\right) & \Leftrightarrow\left(P\land Q\right)\lor\left(P\land R\right)\\ & \Leftarrow P\land Q \end{align*}

(10)

\begin{align*} P\land\left(Q\land R\right) & \Leftrightarrow P\land Q\land R\\ & \Rightarrow P\land Q \end{align*}

(11)

\begin{align*} P\land\left(Q\rightarrow R\right) & \Leftrightarrow P\land\left(\lnot Q\lor R\right)\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\left(P\land R\right)\\ & \Leftarrow P\nrightarrow Q \end{align*}

(12)

\begin{align*} P\land\left(Q\leftarrow R\right) & \Leftrightarrow P\land\left(Q\lor\lnot R\right)\\ & \Leftrightarrow\left(P\land Q\right)\lor\left(P\land\lnot R\right)\\ & \Leftarrow P\land Q \end{align*}

(13)

\begin{align*} P\land\left(Q\downarrow R\right) & \Leftrightarrow P\land\lnot Q\land\lnot R\\ & \Rightarrow P\nrightarrow Q \end{align*}

(14)

\begin{align*} P\land\left(Q\uparrow R\right) & \Leftrightarrow P\land\left(\lnot Q\lor\lnot R\right)\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\left(P\land\lnot R\right)\\ & \Leftarrow P\nrightarrow Q \end{align*}

(15)

\begin{align*} P\land\left(Q\nrightarrow R\right) & \Leftrightarrow P\land Q\land\lnot R\\ & \Rightarrow P\land Q \end{align*}

(16)

\begin{align*} P\land\left(Q\nleftarrow R\right) & \Leftrightarrow P\land\lnot Q\land R\\ & \Rightarrow P\nrightarrow Q \end{align*}

(17)

\begin{align*} P\rightarrow\left(Q\lor R\right) & \Leftrightarrow\lnot P\lor Q\lor R\\ & \Leftarrow P\rightarrow Q \end{align*}

(18)

\begin{align*} P\rightarrow\left(Q\land R\right) & \Leftrightarrow\lnot P\lor\left(Q\land R\right)\\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot P\lor R\right)\\ & \Rightarrow P\rightarrow Q \end{align*}

(19)

\begin{align*} P\rightarrow\left(Q\rightarrow R\right) & \Leftrightarrow\lnot P\lor\lnot Q\lor R\\ & \Leftarrow P\uparrow Q \end{align*}

(20)

\begin{align*} P\rightarrow\left(Q\leftarrow R\right) & \Leftrightarrow\lnot P\lor Q\lor\lnot R\\ & \Leftarrow P\rightarrow Q \end{align*}

(21)

\begin{align*} P\rightarrow\left(Q\downarrow R\right) & \Leftrightarrow\lnot P\lor\left(\lnot Q\land\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left(\lnot P\lor\lnot R\right)\\ & \Rightarrow P\uparrow Q \end{align*}

(22)

\begin{align*} P\rightarrow\left(Q\uparrow R\right) & \Leftrightarrow\lnot P\lor\lnot Q\lor\lnot R\\ & \Leftarrow P\uparrow Q \end{align*}

(23)

\begin{align*} P\rightarrow\left(Q\nrightarrow R\right) & \Leftrightarrow\lnot P\lor\left(Q\land\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot P\lor\lnot R\right)\\ & \Rightarrow P\rightarrow Q \end{align*}

(24)

\begin{align*} P\rightarrow\left(Q\nleftarrow R\right) & \Leftrightarrow\lnot P\lor\left(\lnot Q\land R\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left(\lnot P\lor R\right)\\ & \Rightarrow P\uparrow Q \end{align*}

(25)

\begin{align*} P\leftarrow\left(Q\lor R\right) & \Leftrightarrow P\lor\lnot\left(Q\lor R\right)\\ & \Leftrightarrow P\lor\left(\lnot Q\land\lnot R\right)\\ & \Leftrightarrow\left(P\lor\lnot Q\right)\land\left(P\lor\lnot R\right)\\ & \Rightarrow P\leftarrow Q \end{align*}

(26)

\begin{align*} P\leftarrow\left(Q\land R\right) & \Leftrightarrow P\lor\lnot\left(Q\land R\right)\\ & \Leftrightarrow P\lor\lnot Q\lor\lnot R\\ & \Leftarrow P\leftarrow Q \end{align*}

(27)

\begin{align*} P\leftarrow\left(Q\rightarrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\rightarrow R\right)\\ & \Leftrightarrow P\lor\left(Q\land\lnot R\right)\\ & \Leftrightarrow\left(P\lor Q\right)\land\left(P\lor\lnot R\right)\\ & \Rightarrow P\lor Q \end{align*}

(28)

\begin{align*} P\leftarrow\left(Q\leftarrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\leftarrow R\right)\\ & \Leftrightarrow P\lor\left(\lnot Q\land R\right)\\ & \Leftrightarrow\left(P\lor\lnot Q\right)\\ & \Rightarrow P\leftarrow Q \end{align*}

(29)

\begin{align*} P\leftarrow\left(Q\downarrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\downarrow R\right)\\ & \Leftrightarrow P\lor Q\lor R\\ & \Leftarrow P\lor Q \end{align*}

(30)

\begin{align*} P\leftarrow\left(Q\uparrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\uparrow R\right)\\ & \Leftrightarrow P\lor\left(Q\land R\right)\\ & \Leftrightarrow\left(P\lor Q\right)\land\left(P\lor R\right)\\ & \Rightarrow P\lor Q \end{align*}

(31)

\begin{align*} P\leftarrow\left(Q\nrightarrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\nrightarrow R\right)\\ & \Leftrightarrow P\lor\lnot Q\lor R\\ & \Leftarrow P\leftarrow Q \end{align*}

(32)

\begin{align*} P\leftarrow\left(Q\nleftarrow R\right) & \Leftrightarrow P\lor\lnot\left(Q\nleftarrow R\right)\\ & \Leftrightarrow P\lor Q\lor\lnot R\\ & \Leftarrow P\lor Q \end{align*}

(33)

\begin{align*} P\downarrow\left(Q\lor R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\lor R\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\land\lnot R\\ & \Rightarrow P\downarrow Q \end{align*}

(34)

\begin{align*} P\downarrow\left(Q\land R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\land R\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot Q\lor\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot Q\right)\lor\left(\lnot P\land\lnot R\right)\\ & \Leftarrow P\downarrow Q \end{align*}

(35)

\begin{align*} P\downarrow\left(Q\rightarrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\rightarrow R\right)\\ & \Leftrightarrow\lnot P\land Q\land\lnot R\\ & \Rightarrow P\nleftarrow Q \end{align*}

(36)

\begin{align*} P\downarrow\left(Q\leftarrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\leftarrow R\right)\\ & \Leftrightarrow\lnot P\land\lnot Q\land R\\ & \Rightarrow P\downarrow Q \end{align*}

(37)

\begin{align*} P\downarrow\left(Q\downarrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\downarrow R\right)\\ & \Leftrightarrow\lnot P\land\left(Q\lor R\right)\\ & \Leftrightarrow\left(\lnot P\land Q\right)\lor\left(\lnot P\land R\right)\\ & \Leftarrow P\nleftarrow Q \end{align*}

(38)

\begin{align*} P\downarrow\left(Q\uparrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\uparrow R\right)\\ & \Leftrightarrow\lnot P\land Q\land R\\ & \Rightarrow P\nleftarrow Q \end{align*}

(39)

\begin{align*} P\downarrow\left(Q\nrightarrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\nrightarrow R\right)\\ & \Leftrightarrow\lnot P\land\left(\lnot Q\lor R\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot Q\right)\lor\left(\lnot P\land R\right)\\ & \Leftarrow P\downarrow Q \end{align*}

(40)

\begin{align*} P\downarrow\left(Q\nleftarrow R\right) & \Leftrightarrow\lnot P\land\lnot\left(Q\nleftarrow R\right)\\ & \Leftrightarrow\lnot P\land\left(Q\lor\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\land Q\right)\lor\left(\lnot P\land\lnot R\right)\\ & \Leftarrow P\nleftarrow Q \end{align*}

(41)

\begin{align*} P\uparrow\left(Q\lor R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\lor R\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot Q\land\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left(\lnot P\lor\lnot R\right)\\ & \Rightarrow P\uparrow Q \end{align*}

(42)

\begin{align*} P\uparrow\left(Q\land R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\land R\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\lor\lnot R\\ & \Leftarrow P\uparrow Q \end{align*}

(43)

\begin{align*} P\uparrow\left(Q\rightarrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\rightarrow R\right)\\ & \Leftrightarrow\lnot P\lor\left(Q\land\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot P\lor\lnot R\right)\\ & \Rightarrow P\rightarrow Q \end{align*}

(44)

\begin{align*} P\uparrow\left(Q\leftarrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\leftarrow R\right)\\ & \Leftrightarrow\lnot P\lor\left(\lnot Q\land R\right)\\ & \Leftrightarrow\left(\lnot P\lor\lnot Q\right)\land\left(\lnot P\lor R\right)\\ & \Rightarrow P\uparrow Q \end{align*}

(45)

\begin{align*} P\uparrow\left(Q\downarrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\downarrow R\right)\\ & \Leftrightarrow\lnot P\lor Q\lor R\\ & \Leftarrow P\rightarrow Q \end{align*}

(46)

\begin{align*} P\uparrow\left(Q\uparrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\uparrow R\right)\\ & \Leftrightarrow\lnot P\lor\left(Q\land R\right)\\ & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot P\lor R\right)\\ & \Rightarrow P\rightarrow Q \end{align*}

(47)

\begin{align*} P\uparrow\left(Q\nrightarrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\nrightarrow R\right)\\ & \Leftrightarrow\lnot P\lor\lnot Q\lor R\\ & \Leftarrow P\uparrow Q \end{align*}

(48)

\begin{align*} P\uparrow\left(Q\nleftarrow R\right) & \Leftrightarrow\lnot P\lor\lnot\left(Q\nleftarrow R\right)\\ & \Leftrightarrow\lnot P\lor Q\lor\lnot R\\ & \Leftarrow P\rightarrow Q \end{align*}

(49)

\begin{align*} P\nrightarrow\left(Q\lor R\right) & \Leftrightarrow P\land\lnot\left(Q\lor R\right)\\ & \Leftrightarrow P\land\lnot Q\land\lnot R\\ & \Rightarrow P\nrightarrow Q \end{align*}

(50)

\begin{align*} P\nrightarrow\left(Q\land R\right) & \Leftrightarrow P\land\lnot\left(Q\land R\right)\\ & \Leftrightarrow P\land\left(\lnot Q\lor\lnot R\right)\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\left(P\land\lnot R\right)\\ & \Leftarrow P\nrightarrow Q \end{align*}

(51)

\begin{align*} P\nrightarrow\left(Q\rightarrow R\right) & \Leftrightarrow P\land\lnot\left(Q\rightarrow R\right)\\ & \Leftrightarrow P\land Q\land\lnot R\\ & \Rightarrow P\land Q \end{align*}

(52)

\begin{align*} P\nrightarrow\left(Q\leftarrow R\right) & \Leftrightarrow P\land\lnot\left(Q\leftarrow R\right)\\ & \Leftrightarrow P\land\lnot Q\land R\\ & \Rightarrow P\nrightarrow Q \end{align*}

(53)

\begin{align*} P\nrightarrow\left(Q\downarrow R\right) & \Leftrightarrow P\land\lnot\left(Q\downarrow R\right)\\ & \Leftrightarrow P\land\left(Q\lor R\right)\\ & \Leftrightarrow\left(P\land Q\right)\lor\left(P\land R\right)\\ & \Leftarrow P\land Q \end{align*}

(54)

\begin{align*} P\nrightarrow\left(Q\uparrow R\right) & \Leftrightarrow P\land\lnot\left(Q\uparrow R\right)\\ & \Leftrightarrow P\land Q\land R\\ & \Rightarrow P\land Q \end{align*}

(55)

\begin{align*} P\nrightarrow\left(Q\nrightarrow R\right) & \Leftrightarrow P\land\lnot\left(Q\nrightarrow R\right)\\ & \Leftrightarrow P\land\left(\lnot Q\lor R\right)\\ & \Leftrightarrow\left(P\land\lnot Q\right)\lor\left(P\land R\right)\\ & \Leftarrow P\nrightarrow Q \end{align*}

(56)

\begin{align*} P\nrightarrow\left(Q\nleftarrow R\right) & \Leftrightarrow P\land\lnot\left(Q\nleftarrow R\right)\\ & \Leftrightarrow P\land\left(Q\lor\lnot R\right)\\ & \Leftrightarrow\left(P\land Q\right)\lor\left(P\land\lnot R\right)\\ & \Leftarrow P\land Q \end{align*}

(57)

\begin{align*} P\nleftarrow\left(Q\lor R\right) & \Leftrightarrow\lnot P\land\left(Q\lor R\right)\\ & \Leftrightarrow\left(\lnot P\land Q\right)\lor\left(\lnot P\land R\right)\\ & \Leftarrow P\nleftarrow Q \end{align*}

(58)

\begin{align*} P\nleftarrow\left(Q\land R\right) & \Leftrightarrow\lnot P\land Q\land R\\ & \Rightarrow P\nleftarrow Q \end{align*}

(59)

\begin{align*} P\nleftarrow\left(Q\rightarrow R\right) & \Leftrightarrow\lnot P\land\left(\lnot Q\lor R\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot Q\right)\lor\left(\lnot P\land R\right)\\ & \Leftarrow P\downarrow Q \end{align*}

(60)

\begin{align*} P\nleftarrow\left(Q\leftarrow R\right) & \Leftrightarrow\lnot P\land\left(Q\lor\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\land Q\right)\lor\left(\lnot P\land\lnot R\right)\\ & \Leftarrow P\nleftarrow Q \end{align*}

(61)

\begin{align*} P\nleftarrow\left(Q\downarrow R\right) & \Leftrightarrow\lnot P\land\lnot Q\land\lnot R\\ & \Rightarrow P\downarrow Q \end{align*}

(62)

\begin{align*} P\nleftarrow\left(Q\uparrow R\right) & \Leftrightarrow\lnot P\land\left(\lnot Q\lor\lnot R\right)\\ & \Leftrightarrow\left(\lnot P\land\lnot Q\right)\lor\left(\lnot P\land\lnot R\right)\\ & \Leftarrow P\downarrow Q \end{align*}