`(p implies q) ^^ (q implies p)` is :

Options

A. `p implies q`
B. `p ltimplies q`
C. `p vv q`
D. q

Answer

To solve the expression `(p implies q) ∧ (q implies p)`, we need to analyze what this expression means and how we can simplify it.

### Step-by-Step Solution:

1. **Understanding the Implication**:
The implication `p implies q` can be expressed in logical terms as `¬p ∨ q`, which means "not p or q". Similarly, `q implies p` can be expressed as `¬q ∨ p`.

2. **Rewriting the Expression**:
Therefore, we can rewrite the original expression:
\[ (p \implies q) \land (q \implies p) = (¬p ∨ q) ∧ (¬q ∨ p) \]

3. **Analyzing the Conjunction**:
The conjunction `A ∧ B` is true if both `A` and `B` are true. Thus, for our expression to be true, both `¬p ∨ q` and `¬q ∨ p` must be true.

4. **Truth Table Method**:
We can create a truth table to analyze the conditions under which both parts of the conjunction are true.

| p | q | p implies q (¬p ∨ q) | q implies p (¬q ∨ p) | (p implies q) ∧ (q implies p) |
|---|---|-----------------------|-----------------------|--------------------------------|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |

5. **Identifying the Conditions**:
From the truth table, we see that `(p implies q) ∧ (q implies p)` is true in two cases:
- When both `p` and `q` are true (T, T).
- When both `p` and `q` are false (F, F).

6. **Conclusion**:
The expression `(p implies q) ∧ (q implies p)` is true if and only if `p` and `q` have the same truth value. This is the definition of a biconditional statement, which can be denoted as:
\[ p \iff q \]
Therefore, the expression `(p implies q) ∧ (q implies p)` is equivalent to `p if and only if q`.

### Final Answer:
The expression `(p implies q) ∧ (q implies p)` is equivalent to `p if and only if q`, denoted as \( p \iff q \).