Question

The logical statement $[\sim (\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)]$ is equivalent to:

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

Answer 1

The Correct Option is A

To solve the given logical statement \([\sim (\sim p \vee q) \vee (p \wedge r) \wedge (\sim q \wedge r)]\), we need to simplify it step-by-step using logical equivalences.

Start by simplifying the inner portion \(\sim (\sim p \vee q)\). According to De Morgan's Laws, this can be expressed as:

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

Substitute this back into the main expression:

\([(p \wedge \sim q) \vee (p \wedge r) \wedge (\sim q \wedge r)]\)

Now, apply distribution laws. Distribute \((p \wedge \sim q)\) over the expression:

\([(p \wedge \sim q) \vee ((p \wedge r) \wedge (\sim q \wedge r))]\)

Simplify the expression by distributing terms:

\([(p \wedge \sim q) \vee (p \wedge r \wedge \sim q \wedge r)]\)

Notice that \((p \wedge r \wedge \sim q \wedge r)\) simplifies to \((p \wedge r \wedge \sim q)\) (since \(r \wedge r = r\)):

\([(p \wedge \sim q) \vee (p \wedge r \wedge \sim q)]\)

Combine the terms using distribution:

\([(p \wedge \sim q)]\) or \([(p \wedge r) \wedge \sim q]\)

The simplified logical expression is equivalent to \((p \wedge r) \wedge \sim q\), which matches the given correct answer.