Chapter 1, Problem 1RE

Construct a truth table for the compound statement $\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$.

To determine

The truth table for the compound statement $\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$.

Answer to Problem 1RE

Solution: The truth table for the compound statement $\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$ is

$p$	$q$	$r$	$\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	T	T
F	T	T	T
T	F	T	T
T	T	F	F
F	F	F	T

Explanation of Solution

Let us draw the truth table for the compound statement $\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$.

$p$	$q$	$r$	$\neg q$	$\neg r$	$q\to \left(\neg r\right)$	$\left(\neg q\right)\vee r$	$p\wedge \left(q\to \left(\neg r\right)\right)$	$\left[p\wedge \left(q\to \left(\neg r\right)\right)\right]\to \left[\left(\neg q\right)\vee r\right]$
T	T	T	F	F	F	T	F	T
T	F	F	T	T	T	T	T	T
F	T	F	F	T	T	F	F	T
F	F	T	T	F	T	T	F	T
F	T	T	F	F	F	T	F	T
T	F	T	T	F	T	T	T	T
T	T	F	F	T	T	F	T	F
F	F	F	T	T	T	T	F	T