Question

construct a truth table for the given statement.
(¬q ∧ p) ∨ (q ∧ p)
complete the truth table.

p	q	¬q	¬q ∧ p	q ∧ p	(¬q ∧ p) ∨ (q ∧ p)
t	t	f	f	t	t
t	f	t	t	f	t
f	t	▼	▼	▼	▼

Explanation:

Step1: Analyze $\sim q$ when $p = F, q = T$

The negation of $q$ (denoted as $\sim q$) is the opposite truth value of $q$. Since $q = T$, then $\sim q = F$.

Step2: Analyze $\sim q \wedge p$ when $p = F, q = T$

The conjunction ($\wedge$) is true only when both components are true. Here, $\sim q = F$ and $p = F$, so $\sim q \wedge p = F \wedge F = F$.

Step3: Analyze $q \wedge p$ when $p = F, q = T$

The conjunction of $q$ and $p$: $q = T$ and $p = F$, so $q \wedge p = T \wedge F = F$.

Step4: Analyze $(\sim q \wedge p) \vee (q \wedge p)$ when $p = F, q = T$

The disjunction ($\vee$) is true if at least one component is true. Here, $\sim q \wedge p = F$ and $q \wedge p = F$, so $(\sim q \wedge p) \vee (q \wedge p) = F \vee F = F$.

Now let's fill the table row by row for the case $p = F, q = T$:

• $\sim q$: $F$ (from Step1)
• $\sim q \wedge p$: $F$ (from Step2)
• $q \wedge p$: $F$ (from Step3)
• $(\sim q \wedge p) \vee (q \wedge p)$: $F$ (from Step4)

Answer:

For the row where $p = F$ and $q = T$:

• $\sim q$: $\boldsymbol{F}$
• $\sim q \wedge p$: $\boldsymbol{F}$
• $q \wedge p$: $\boldsymbol{F}$
• $(\sim q \wedge p) \vee (q \wedge p)$: $\boldsymbol{F}$

The completed row (third row) is:

$p$	$q$	$\sim q$	$\sim q \wedge p$	$q \wedge p$	$(\sim q \wedge p) \vee (q \wedge p)$