Question

\([(p\wedge q)\vee (\sim r\vee \sim s\vee \sim t)]\wedge [(p\wedge q)\vee (r\wedge s\wedge t)]\equiv\)

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

Hint:

Rewrite the negated part as the negation of a single conjunction, then use the distributive law.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let \(A=p\wedge q\) and \(B=r\wedge s\wedge t\).
By De Morgan's law, \(\sim r\vee \sim s\vee \sim t\equiv \sim(r\wedge s\wedge t)=\sim B\).

Step 2: Rewrite the Statement
The given statement becomes \((A\vee \sim B)\wedge (A\vee B)\).

Step 3: Apply the Distributive Law
\((A\vee \sim B)\wedge (A\vee B)\equiv A\vee (\sim B\wedge B)\).
Since \(\sim B\wedge B\) is always false, the expression reduces to \(A\vee F\equiv A\).

Step 4: Final Result
So the statement is equivalent to \(A=p\wedge q\), which matches option (C).

Final Answer:
The statement is equivalent to \(p\wedge q\). \[ \boxed{\text{(C) }p\wedge q} \]