Question

If $p$, $p\Rightarrow q$ and $\sim r \Rightarrow q$ are true, then consider the following statements

(i) $r$ is true (ii) $r$ is false (iii) $q$ is false (iv) $p \lor q$ is false

Answer 1

None of the above statements are necessarily true.

Answer 2

We are given that:

1. $p$ is true.
2. $p \Rightarrow q$ is true.
3. $\sim r \Rightarrow q$ is true.

Step 1: Since $p$ is true and $p \Rightarrow q$ is true, we must have $q$ true.

Step 2: With $q$ true, the implication $\sim r \Rightarrow q$ is automatically true regardless of the truth value of $r$. Therefore, $r$ can be either true or false.

Step 3: Evaluate each statement:

• (i) " $r$ is true": Not necessarily; $r$ may be true or false.
• (ii) " $r$ is false": Not necessarily; same reason as above.
• (iii) " $q$ is false": False, because we derived $q$ is true.
• (iv) " $p \lor q$ is false": False, because both $p$ and $q$ are true, hence $p \lor q$ is true.

Conclusion: None of the statements (i), (ii), (iii), or (iv) must be true.