Question

If $p, q$ are true statements and $r$ is false statement, then which of the following is correct.

• A. $(p \vee q) \vee r$ has truth value F
• B. $(p \wedge q) \rightarrow r$ has truth value F
• C. $(p \rightarrow r) \rightarrow q$ has truth value T
• D. $(p \leftrightarrow q) \rightarrow r$ has truth value F

Hint:

For implication $P \rightarrow Q$, the statement is only False if $P$ is True and $Q$ is False.

Answer 1

The Correct Option is D

Step 1: Note the truth values.
We are told $p$ and $q$ are true and $r$ is false. We test the statement in option 4.

Step 2: Inner biconditional.
Since $p$ and $q$ are both true, $p \leftrightarrow q$ is true (both sides agree).

Step 3: The implication.
Now $(p\leftrightarrow q)\rightarrow r$ becomes $\text{T}\rightarrow \text{F}$. A true statement implying a false one is false.

Step 4: Match the claim.
Option 4 says this has truth value F, which is exactly what we found, so option 4 is the correct choice. \[ \boxed{(p\leftrightarrow q)\rightarrow r \text{ is } F} \]