Question:medium

Identify the truth value of the statement pattern \( (p \land \sim q) \rightarrow r \).

Show Hint

An implication \(p \rightarrow q\) is false only when \(p\) is true and \(q\) is false; in all other cases it is true.
Updated On: Apr 15, 2026
  • Tautology
  • Contradiction
  • Contingency
  • None of these
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Understanding the Question:
The question asks to classify a logical expression based on its truth table.
A Tautology is always True, a Contradiction is always False, and a Contingency is a mix of both.
Step 2: Key Formula or Approach:
Use a Truth Table to evaluate all possible combinations of truth values for \( p \), \( q \), and \( r \).
The implication \( X \to Y \) is false only when \( X \) is True and \( Y \) is False.
Step 3: Detailed Explanation:
We evaluate the expression \( (p \land \sim q) \to r \):
1. If \( p=T, q=F, r=F \):
\( \sim q = T \).
\( (p \land \sim q) = (T \land T) = T \).
\( T \to F = F \).
2. If \( p=T, q=T, r=T \):
\( \sim q = F \).
\( (p \land \sim q) = (T \land F) = F \).
\( F \to T = T \).
Since we found at least one case where the statement is False and at least one case where it is True, the statement is neither a tautology nor a contradiction.
Step 4: Final Answer:
The statement pattern is a Contingency.
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