The question asks to evaluate the truth value of the following logical statement:

$(p \iff q) \land (q \iff r)$

Given the conditions:

• $p$ is true,
• $q$ is false,
• $r$ is false.

Step 1: Evaluate $p \iff q$

The biconditional ($\iff$) is true when both sides have the same truth value.

• $p = \text{true}$,
• $q = \text{false}$.

Since $p$ and $q$ have different truth values, $p \iff q = \text{false}$.

Step 2: Evaluate $q \iff r$

• $q = \text{false}$,
• $r = \text{false}$.

Since both $q$ and $r$ are false, $q \iff r = \text{true}$ (because both sides have the same truth value).

Step 3: Evaluate the entire expression

The overall expression is $(p \iff q) \land (q \iff r)$.

• $p \iff q = \text{false}$,
• $q \iff r = \text{true}$.

The conjunction ($\land$) is true only if both statements are true. Since one of them is false, the entire expression evaluates to false.

Final Answer:

The truth value of the given expression is false.

Would you like more details or have any other questions?

Here are 5 related questions:

1. What is the truth table for the biconditional ($\iff$) operator?
2. How does conjunction ($\land$) affect