Question

Which one of the following is false? Read $\wedge$ as AND , $\vee$ as OR, $\sim$ as NOT, $\to$ as one way implication and $\leftrightarrow$ as two way implication .

• A. $((x\to y)\wedge x)\to y$
• B. $((\sim x\to y)\wedge (\sim x\to \sim y))\to x$
• C. $(x\to (x\vee y))$
• D. $((x\vee y)\leftrightarrow (\sim x\to \sim y))$

Answer 1

The correct option is D $((x\vee y)\leftrightarrow (\sim x\to \sim y))$

(a)
$((x\to y)\wedge x)\to y$
$=\sim \left(\right(\sim x\vee y)\wedge x)\vee y$
$=\sim \left(\right(\sim x\wedge x)\vee (y\wedge x\left)\right)\vee y$
$=\sim (F\vee (y\wedge x\left)\right)\vee y$
$=\sim (y\wedge x))\vee y$
$=\sim y\vee \sim x\vee y$
$=(\sim y\vee y)\vee \sim x$
$=T\vee \sim x$
=T
(b)
$((\sim x\to y)\wedge (\sim x\to \sim y\left)\right)\to x$
$=\sim (\sim (\sim x)\wedge y)\wedge (\sim (\sim x)\vee \sim y))\vee x$
$=\sim \left(\right(\sim x\vee y)\wedge (x\vee \sim y\left)\right)\vee x$
$=\sim (x\vee (y\wedge \sim y\left)\right)\vee x$
$=\sim (x\vee F)\vee x$
$=\sim x\vee x$
= T
(c)
$(x\to (x\vee y\left)\right)$
= $(\sim x\vee (x\vee y\left)\right)$
= $\left(\right(\sim x\vee x)\vee y)$
= $(T\vee y)$
= T

(d)
$\left(\right(x\vee y)\leftrightarrow (\sim x\to \sim y\left)\right)$
= $(x\vee y)\leftrightarrow (\sim (\sim x)\vee \sim y$
= $(x\vee y)\leftrightarrow (x\vee \sim y$ )
$=\left(\right(x\vee y)\wedge (x\vee \sim y\left)\right)\vee (\sim (x\vee y)\wedge \sim (x\vee \sim y\left)\right)$
$=\left(\right(x\vee (y\wedge \sim y))\vee ((\sim x\wedge \sim y)\wedge \sim (x\vee \sim y))$
$=(x\vee F)\vee \left(\right(\sim x\wedge \sim y)\wedge (\sim x\wedge y\left)\right)$
$=x\vee (\sim x\wedge (y\wedge \sim y\left)\right)=x\vee (\sim x\wedge y)$
= x $\vee$ F = x