Question

Let $p$ and $q$ be two statements Then $\sim(p \wedge(p \Rightarrow \sim q))$ is equivalent to

• A. $p \vee(p \wedge q)$
• B. $(\sim p) \vee q$
• C. $p \vee(p \wedge(\sim q))$
• D. $p \vee((\sim p) \wedge q)$

Answer 1

The Correct Option is B

To solve the given logical expression \(\sim(p \wedge (p \Rightarrow \sim q))\) and simplify it to find its equivalent, we will proceed step-by-step.

Step 1: Understand the Implication

The logical implication \(p \Rightarrow \sim q\) can be rewritten using its equivalent form: \(\sim p \vee \sim q\).

Step 2: Substitute the Implication

Substitute the implication in the expression:

p \wedge (p \Rightarrow \sim q) = p \wedge (\sim p \vee \sim q)

Step 3: Apply Distributive Law

Apply the distributive law of logic to the expression:

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

Here, p \wedge \sim p is always false, because a statement and its negation cannot both be true.

Step 4: Simplify the Expression

Since p \wedge \sim p is false (denoted by F or 0 in Boolean algebra), the expression simplifies to:

= F \vee (p \wedge \sim q)

= p \wedge \sim q

Step 5: Apply Negation

Now, apply the negation to the entire expression:

\sim (p \wedge \sim q) can be simplified using De Morgan's laws:

= \sim p \vee q

Conclusion

Thus, the expression \sim(p \wedge(p \Rightarrow \sim q)) simplifies to (\sim p) \vee q, which matches the given correct answer.