`(p implies q) hArr` .........

Answer

To solve the question `(p implies q) hArr`, we need to determine the logical equivalence of the statement "P implies Q".

### Step-by-Step Solution:

1. **Understanding the Implication**:
The implication "P implies Q" (denoted as \( P \rightarrow Q \)) can be understood as: if P is true, then Q must also be true. If P is false, the implication is true regardless of the truth value of Q.

2. **Using Logical Equivalence**:
The logical equivalence of \( P \rightarrow Q \) can be expressed using disjunction (OR) and negation (NOT). The equivalence is given by:
\[ P \rightarrow Q \equiv \neg P \lor Q \]
This means that "P implies Q" is logically equivalent to "not P or Q".

3. **Constructing the Truth Table**:
To verify this equivalence, we can construct a truth table for both \( P \rightarrow Q \) and \( \neg P \lor Q \).

| P | Q | \( P \rightarrow Q \) | \( \neg P \) | \( \neg P \lor Q \) |
|---|---|---------------------|-----------|-------------------|
| T | T | T | F | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

From the table, we can see that the columns for \( P \rightarrow Q \) and \( \neg P \lor Q \) are identical, confirming our equivalence.

4. **Conclusion**:
Therefore, we conclude that:
\[ P \rightarrow Q \equiv \neg P \lor Q \]

### Final Answer:
The equivalence of \( P \rightarrow Q \) is \( \neg P \lor Q \).

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