Question

If the truth value of the compound statement \([(p\leftrightarrow q)∧(q\rightarrow r)∧\sim r]\rightarrow (p∧\sim q)\) is false, then the truth values of the statement patterns \((p\rightarrow q)\leftrightarrow (q\rightarrow r)\) and \(\sim (p∨r)\rightarrow (q∧p)\) are, respectively ...

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

Hint:

A conditional is false only when the antecedent is true and the consequent is false.

Answer 1

The Correct Option is B

Step 1: Work backwards
For the whole statement to be false we need the premise true and $p\wedge\sim q$ false. The premise has three parts, each of which must be true: $p\leftrightarrow q$, $q\rightarrow r$, $\sim r$.

Step 2: Assign
$\sim r = T$ gives $r=F$. $q\rightarrow F$ true gives $q=F$. $p\leftrightarrow F$ true gives $p=F$.

Step 3: Evaluate
Pattern 1: $(F\rightarrow F)\leftrightarrow(F\rightarrow F) = T\leftrightarrow T = T$. Pattern 2: $\sim(F\vee F)\rightarrow(F\wedge F) = T\rightarrow F = F$. Option (B).

Final Answer:
(T, F). \[ \boxed{\text{(B)}\ (T,F)} \]