QJEEM23Mac8ce: JEE Main 2023 Mathematics - Class 11 JEEM23Mac8ce Solution: Negation of is :

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:

1. $\sim (A \wedge B) \equiv \sim A \vee \sim B$
2. $\sim (A \vee B) \equiv \sim A \wedge \sim B$

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 ∨ ¬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 :

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