Question

If \(\sim p\rightarrow q\) is false and \(q\leftrightarrow r\) is false, then the truth value of \((p,q,r)\) is...

• A. \((T,F,T)\)
• B. \((F,T,F)\)
• C. \((F,F,T)\)
• D. \((F,T,T)\)

Hint:

A false conditional means the antecedent is true and the consequent is false.

Answer 1

The Correct Option is C

Step 1: Use the first statement
\(\sim p \to q\) is false only if \(\sim p\) is true and \(q\) is false. So \(p = F\) and \(q = F\).

Step 2: Use the second statement
\(q \leftrightarrow r\) is false when \(q\) and \(r\) have different truth values. Since \(q = F\), we need \(r = T\).

Step 3: Combine
\((p, q, r) = (F, F, T)\).

Step 4: Check the options
(A) \((T,F,T)\) has \(p = T\), so \(\sim p \to q\) would be true. (B) \((F,T,F)\) has \(q = T\), making \(\sim p \to q\) true. (D) \((F,T,T)\) also has \(q = T\). Only (C) satisfies both conditions.

Final Answer:
The values are (F, F, T). This is option (C). \[ \boxed{\text{(C) }(F,F,T)} \]