Question

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

• A. Tautology
• B. Contradiction
• C. Contingency
• D. None of these

Hint:

An implication \(p \rightarrow q\) is false only when \(p\) is true and \(q\) is false; in all other cases it is true.

Answer 1

The Correct Option is C

Concept: In propositional logic:

• Tautology: A statement that is always true.
• Contradiction: A statement that is always false.
• Contingency: A statement that is sometimes true and sometimes false.

The implication \( p \rightarrow q \) is false only when \( p \) is true and \( q \) is false.
Step 1: {Construct the truth table.} \[ \begin{array}{c|c|c|c|c} p & q & r & p \land \sim q & (p \land \sim q) \rightarrow r
\hline T & T & T & F & T
T & T & F & F & T
T & F & T & T & T
T & F & F & T & F
F & T & T & F & T
F & T & F & F & T
F & F & T & F & T
F & F & F & F & T \end{array} \]
Step 2: {Analyze the final column.} The final column contains both \(T\) and \(F\). Thus, the statement is sometimes true and sometimes false. \[ \therefore \text{The statement is a Contingency.} \]