The statement among the following that is a tautology is :

Options

A. ` A vv (A ^^ B)`
B. `A ^^ (A vv B)`
C. `B to [A ^^ (A to B)]`
D. `[A ^^ (A to B)] to B `

Answer

To determine which of the given statements is a tautology, we will analyze each option using truth tables. A tautology is a statement that is always true regardless of the truth values of its components.

Let's denote the options as follows:
1. \( A \cup (A \cap B) \)
2. \( A \cap (A \cup B) \)
3. \( A \Rightarrow B \)
4. \( A \Leftrightarrow A \)

We'll evaluate each option step by step.

### Step 1: Evaluate Option 1 - \( A \cup (A \cap B) \)

1. **Construct the truth table for \( A \) and \( B \)**:
- \( A \): True (T), True (T), False (F), False (F)
- \( B \): True (T), False (F), True (T), False (F)

| A | B | A ∩ B | A ∪ (A ∩ B) |
|-------|-------|-------|-------------|
| T | T | T | T |
| T | F | F | T |
| F | T | F | F |
| F | F | F | F |

- The column for \( A \cup (A \cap B) \) shows T, T, F, F. This is not a tautology.

### Step 2: Evaluate Option 2 - \( A \cap (A \cup B) \)

1. **Construct the truth table for \( A \) and \( B \)**:

| A | B | A ∪ B | A ∩ (A ∪ B) |
|-------|-------|-------|-------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | F |
| F | F | F | F |

- The column for \( A \cap (A \cup B) \) shows T, T, F, F. This is not a tautology.

### Step 3: Evaluate Option 3 - \( A \Rightarrow B \)

1. **Construct the truth table for \( A \) and \( B \)**:

| A | B | A ⇒ B |
|-------|-------|-------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |

- The column for \( A \Rightarrow B \) shows T, F, T, T. This is not a tautology.

### Step 4: Evaluate Option 4 - \( A \Leftrightarrow A \)

1. **Construct the truth table for \( A \)**:

| A | A ⇔ A |
|-------|-------|
| T | T |
| F | T |

- The column for \( A \Leftrightarrow A \) shows T, T. This is always true, hence it is a tautology.

### Conclusion

The statement that is a tautology among the given options is \( A \Leftrightarrow A \).