Question

The logical statement \((p∨q)∧[(\sim p∧q)∨(p∧\sim q)]∧\sim q\) is logically equivalent to ...

• A. \(p∧\sim q\)
• B. \(\sim p∧q\)
• C. \(p∧q\)
• D. \(\sim p∨\sim q\)

Hint:

Use the exclusive-or shape of the middle bracket and the final ~q.

Answer 1

The Correct Option is A

Step 1: Truth Table:
Check the four rows. TT: $\sim q$ false, so the statement is F. FT: $\sim q$ false, so F. TF: $p\vee q=T$, the XOR is T, $\sim q=T$, so T. FF: $p\vee q=F$, so F.

Step 2: Compare:
The statement is true only when $p=T,\ q=F$. That is exactly the truth table of $p\wedge\sim q$.

Step 3: Check Others:
$\sim p\wedge q$ is true only for FT, $p\wedge q$ only for TT, and $\sim p\vee\sim q$ is true in three rows. Option (A).

Final Answer:
Option (A). \[ \boxed{\text{(A) } p\wedge\sim q} \]