Negation of p (q (p q)) is :

Question

Negation of $p \wedge(q \wedge \sim(p \wedge q))$ is :

Accepted Answer

The given statement is:

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

We want to find the negation of this statement. To do that, we negate the whole expression:

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

Now, we will use De Morgan's laws to simplify the expression. De Morgan's laws state:

$\sim (A \wedge B) \equiv \sim A \vee \sim B$
$\sim (A \vee B) \equiv \sim A \wedge \sim B$

Applying the first De Morgan's law to the outer conjunction of the negated expression:

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

Now we need to address the inner conjunction $(q \wedge \sim (p \wedge q))$. To simplify this part of the expression, we again apply the first De Morgan's law:

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

Distribute the disjunction over the inner conjunction:

$$
\sim \mathrm{p} \vee((\sim \mathrm{q} \vee \mathrm{p}) \wedge(\sim \mathrm{q} \vee \mathrm{q}))
$$

Apply the law of excluded middle (A \vee  \neg A = T) on the last term:

$$
\sim \mathrm{p} \vee(\sim \mathrm{q} \vee \mathrm{p})
$$

Apply De Morgan's law on the inner disjunction:

$$
\sim(\mathrm{p} \wedge \mathrm{q}) \vee \mathrm{p}
$$

Now, comparing this with the provided options, it matches with Option B :

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

So, the negation of the given statement is :

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

Solutions · 1