Question

construct the truth table for the compound statement (p ∧ ~q) ∨ r. fill in the empty values of the truth table. start with the first four rows now fill in the final four rows

Explanation:

Step1: Recall logical - operator rules

The negation $\sim q$ is false when $q$ is true and true when $q$ is false. The conjunction $p\land\sim q$ is true when both $p$ is true and $\sim q$ is true. The disjunction $(p\land\sim q)\lor r$ is true when either $p\land\sim q$ is true or $r$ is true.

Step2: Analyze first - four rows

Row 1: $p = T,q = T,r = T$

$\sim q=F$, $p\land\sim q = F$, $(p\land\sim q)\lor r=T$

Row 2: $p = T,q = T,r = F$

$\sim q=F$, $p\land\sim q = F$, $(p\land\sim q)\lor r=F$

Row 3: $p = T,q = F,r = T$

$\sim q=T$, $p\land\sim q = T$, $(p\land\sim q)\lor r=T$

Row 4: $p = T,q = F,r = F$

$\sim q=T$, $p\land\sim q = T$, $(p\land\sim q)\lor r=T$

Step3: Analyze last four rows

Row 5: $p = F,q = T,r = T$

$\sim q=F$, $p\land\sim q = F$, $(p\land\sim q)\lor r=T$

Row 6: $p = F,q = T,r = F$

$\sim q=F$, $p\land\sim q = F$, $(p\land\sim q)\lor r=F$

Row 7: $p = F,q = F,r = T$

$\sim q=T$, $p\land\sim q = F$, $(p\land\sim q)\lor r=T$

Row 8: $p = F,q = F,r = F$

$\sim q=T$, $p\land\sim q = F$, $(p\land\sim q)\lor r=F$

Answer:

$p$	$q$	$r$	$(p\land\sim q)\lor r$
$T$	$T$	$F$	$F$
$T$	$F$	$T$	$T$
$T$	$F$	$F$	$T$
$F$	$T$	$T$	$T$
$F$	$T$	$F$	$F$
$F$	$F$	$T$	$T$
$F$	$F$	$F$	$F$