Question

\(\begin{array}{l}(p \land r )\Leftrightarrow ( p \land (\sim q))\end{array}\)is equivalent to (~ p) when r is

• A. p
• B. ∼p
• C. q
• D. ∼q

Answer 1

The Correct Option is C

Let's solve the logical equivalence problem given in the question.

1. We need to determine for which value of r the expression \((p \land r) \Leftrightarrow (p \land (\sim q))\) is equivalent to \((\sim p)\).
2. Start by analyzing the logical expressions:
   • \(p \land r\) means both p and r need to be true. 
   • \(p \land (\sim q)\) means p is true and q is false.
   •  
   • Consider the possible scenarios:
     • If p is false, both \((p \land r)\) and \((p \land (\sim q))\) become false, satisfying the condition \((p \land r) \Leftrightarrow (p \land (\sim q))\).
     • Thus,

Answer 2

The truth table

Clearly \(p∧q⇔p∧∼q≡∼p\)
\(∴ r = q\)