Question

The logical expression \([\sim (p\rightarrow q)\wedge (q\rightarrow p)]\vee \sim (p\leftrightarrow q)\) is equivalent to:

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

Hint:

Convert each implication and biconditional into AND, OR and NOT, then simplify the first bracket before joining it with the second part.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Rewrite the connectives using AND, OR and NOT.
\(p\rightarrow q\equiv \sim p\vee q\) and \(p\leftrightarrow q\equiv (p\rightarrow q)\wedge (q\rightarrow p)\).

Step 2: Simplify the first bracket
\(\sim (p\rightarrow q)\equiv p\wedge \sim q\) and \(q\rightarrow p\equiv \sim q\vee p\).
Their conjunction is \((p\wedge \sim q)\wedge (p\vee \sim q)\equiv p\wedge \sim q\).

Step 3: Join with the second part
\(\sim (p\leftrightarrow q)\equiv (p\wedge \sim q)\vee (\sim p\wedge q)\).
So the whole expression becomes \((p\wedge \sim q)\vee (\sim p\wedge q)\).

Step 4: Compare the options
This is true exactly when one of \(p,q\) is true and not both, which is \((p\vee q)\wedge \sim (p\wedge q)\).

Final Answer:
The expression is equivalent to the exclusive OR, \((p\vee q)\wedge \sim (p\wedge q)\). \[ \boxed{\text{(B) }(p\vee q)\wedge \sim (p\wedge q)} \]