3つのうち1つを消したものとの包含関係
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。
\(P,Q,R\)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*}(63)
\begin{align*} P\nleftarrow\left(Q\nrightarrow R\right) & \Leftrightarrow\lnot P\land Q\land\lnot R\\ & \Rightarrow P\nleftarrow Q \end{align*}(64)
\begin{align*} P\nleftarrow\left(Q\nleftarrow R\right) & \Leftrightarrow\lnot P\land\lnot Q\land R\\ & \Rightarrow P\downarrow Q \end{align*}ページ情報
| タイトル | 3つのうち1つを消したものとの包含関係 |
| URL | https://www.nomuramath.com/r14f2zuh/ |
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存在命題(論理和)と全称命題(論理積)の順序変更
\[
\exists x\in X,\forall y\in Y,P\left(x,y\right)\Rightarrow\forall y\in Y,\exists x\in X,P\left(x,y\right)
\]
量化子(全称命題・存在命題)と空集合
\[
\forall x\in\emptyset,P\left(x\right)\Leftrightarrow\top
\]
逆・裏・対偶の定義と対偶の法則
\[
\left(P\rightarrow Q\right)\Leftrightarrow\left(\lnot P\leftarrow\lnot Q\right)
\]
論理演算子の移項
\[
\left(P\land R\right)\rightarrow Q\Leftrightarrow P\rightarrow\left(Q\lor\lnot R\right)
\]