Question

Consider the following statements
\(r\): If \(p\rightarrow q\) is false then \(p∨q\) is false.
\(s\): If \(p\leftrightarrow q\) is false then \(p∨q\) is false.
The truth values of \(r\rightarrow s\) and \(s\rightarrow r\) are respectively ____

• A. T, T
• B. T, F
• C. F, T
• D. F, F

Hint:

Find the truth values of r and s by testing the antecedent cases.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
\(r\) and \(s\) are themselves implications. We must first decide if each is true or false for all values of \(p\), \(q\).

Step 2: Check r:
\(p\rightarrow q\) is false only when \(p=T,\ q=F\). Then \(p\vee q=T\), not false. So the implication "false \(\Rightarrow\) false" fails. Statement \(r\) is FALSE.

Step 3: Check s:
\(p\leftrightarrow q\) is false when \(p\) and \(q\) differ, for example \(p=T,\ q=F\). Then \(p\vee q=T\), not false. Statement \(s\) is FALSE.

Step 4: Find the truth values of the compound statements:
\(r\rightarrow s\): F \(\rightarrow\) F is T. \(s\rightarrow r\): F \(\rightarrow\) F is T. So the answers are T, T, option A.

Final Answer:
Both implications are true. \[ \boxed{\text{(A) }\text{T, T}} \]