If p is any logical statement, then

Options

A. `p ^^ (~p)` is a tautology
B. `p vv (~p)` is a contradiction
C. `p ^^ p = p`
D. `p vv (~p) = p`

Answer

To solve the question "If p is any logical statement, then", we need to analyze the logical operations involving the statement p and its negation (¬p). We'll create a truth table to evaluate the expressions provided in the options.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \( p \) be a logical statement which can either be True (T) or False (F).
- The negation of \( p \) is denoted as \( ¬p \).

2. **Construct the Truth Table**:
We will create a truth table with the following columns:
- Column 1: \( p \)
- Column 2: \( ¬p \)
- Column 3: \( p \land ¬p \) (AND operation)
- Column 4: \( p \lor ¬p \) (OR operation)
- Column 5: \( p \land p \) (AND operation)
- Column 6: \( p \lor p \) (OR operation)

The truth table will look like this:

| \( p \) | \( ¬p \) | \( p \land ¬p \) | \( p \lor ¬p \) | \( p \land p \) | \( p \lor p \) |
|---------|----------|-------------------|------------------|------------------|-----------------|
| T | F | F | T | T | T |
| F | T | F | T | F | F |

3. **Evaluate Each Expression**:
- \( p \land ¬p \): This expression is always False (F) since a statement cannot be both True and False at the same time.
- \( p \lor ¬p \): This expression is always True (T) because either \( p \) is True or its negation \( ¬p \) is True.
- \( p \land p \): This expression is equivalent to \( p \) itself, so it returns the same value as \( p \).
- \( p \lor p \): This expression is also equivalent to \( p \) itself, returning the same value as \( p \).

4. **Identify the Correct Options**:
- \( p \land ¬p \) is not a tautology (not always True).
- \( p \lor ¬p \) is not a contradiction (not always False).
- \( p \land p = p \) is correct.
- \( p \lor ¬p = p \) is incorrect.

Thus, the correct conclusion is that the expression \( p \land p \) is equivalent to \( p \).

### Final Answer:
The correct option is \( p \land p = p \).