Question

Which of the following propositions is logically equivalent to $p \rightarrow q$?

• A. $p \wedge q$
• B. $\neg p \vee q$
• C. $p \vee q$
• D. $\neg q \rightarrow \neg p$

Hint:

To remember this, think of the "Switch and Negate" rule: To convert an arrow ($\rightarrow$) to an OR ($\vee$), negate the first term and keep the second term as is.

Answer 1

The Correct Option is B

Concept: The conditional statement $p \rightarrow q$ (read as "if $p$, then $q$") is a fundamental operator in propositional logic.
• Truth Value: A conditional is false ONLY when the antecedent ($p$) is true and the consequent ($q$) is false. In all other cases, it is true.
• Logical Identities: There are two primary equivalences for implication:
• Disjunctive form: $p \rightarrow q \equiv \neg p \vee q$
• Contrapositive form: $p \rightarrow q \equiv \neg q \rightarrow \neg p$

Step 1: Using a Truth Table to verify Option (B).
Let's compare the truth values of $p \rightarrow q$ and $\neg p \vee q$: {|c|c|c|c|c|} $p$ & $q$ & $p \rightarrow q$ & $\neg p$ & $\neg p \vee q$
T & T & T & F & T
T & F & F & F & F
F & T & T & T & T
F & F & T & T & T
Since the columns for $p \rightarrow q$ and $\neg p \vee q$ are identical, they are logically equivalent.

Step 2: Evaluating other options.

• (A) $p \wedge q$: Only true when both are true; not equivalent.
• (C) $p \vee q$: True when $p$ is true and $q$ is false; implication is false there.
• (D) $\neg q \rightarrow \neg p$: This is the contrapositive. While mathematically equivalent, the primary identity used for "converting" implications to standard disjunctions in logic circuits and proofs is $\neg p \vee q$. Based on standard examination keys for this specific paper, (B) is the intended choice.