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: Work through the first clue
For $\sim p \to q$ to fail we need $\sim p = T$ and $q = F$, so $p = F$ and $q = F$.

Step 2: Second clue
$q \leftrightarrow r$ false means $q \neq r$ in value, so $r = T$.

Step 3: Verify
With $(F, F, T)$: $\sim p = T$, $T \to F = F$ (given). $F \leftrightarrow T = F$ (given). Both hold.

Step 4: Answer
Option (C), $(F, F, T)$.

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