Write 'T' for True and 'F' for False.
`(p hArr q) hArr (p implies q) ^^ (q implies p)`.

Answer

To determine the truth value of the statement \((p \iff q) \iff (p \implies q) \land (q \implies p)\), we will analyze both sides of the biconditional statement.

### Step 1: Understand the components of the statement
The statement consists of two main parts:
1. \(p \iff q\) (p if and only if q)
2. \((p \implies q) \land (q \implies p)\) (p implies q and q implies p)

### Step 2: Analyze \(p \iff q\)
The biconditional \(p \iff q\) is true if both \(p\) and \(q\) are either true or false. In other words:
- If \(p\) is true, then \(q\) must be true.
- If \(p\) is false, then \(q\) must be false.

### Step 3: Analyze \((p \implies q) \land (q \implies p)\)
The expression \((p \implies q) \land (q \implies p)\) means:
- \(p \implies q\) is true if whenever \(p\) is true, \(q\) is also true.
- \(q \implies p\) is true if whenever \(q\) is true, \(p\) is also true.

This means that both implications must hold true simultaneously for the conjunction to be true.

### Step 4: Compare both sides
Now, we need to see if both sides of the biconditional are equivalent.

- If \(p\) and \(q\) are both true, then:
- \(p \iff q\) is true.
- \(p \implies q\) is true and \(q \implies p\) is true, hence \((p \implies q) \land (q \implies p)\) is true.

- If \(p\) and \(q\) are both false, then:
- \(p \iff q\) is true.
- \(p \implies q\) is true (since false implies anything) and \(q \implies p\) is true (for the same reason), hence \((p \implies q) \land (q \implies p)\) is true.

- If \(p\) is true and \(q\) is false, then:
- \(p \iff q\) is false.
- \(p \implies q\) is false (true does not imply false) and \(q \implies p\) is true, hence \((p \implies q) \land (q \implies p)\) is false.

- If \(p\) is false and \(q\) is true, then:
- \(p \iff q\) is false.
- \(p \implies q\) is true (false implies anything) and \(q \implies p\) is false, hence \((p \implies q) \land (q \implies p)\) is false.

### Step 5: Conclusion
In all cases, the truth values of both sides of the biconditional are the same. Therefore, the entire statement \((p \iff q) \iff (p \implies q) \land (q \implies p)\) is true.

Thus, the answer is:
**T** (True)