The only statement among the following i.e. a tautology is

Options

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

Explanation

To determine which of the given statements is a tautology, we will analyze each option step by step using logical reasoning and truth tables.

### Step 1: Understand Tautology
A tautology is a statement that is always true regardless of the truth values of its components.

### Step 2: Analyze the Options
Let's denote the options as follows:
1. Option 1: \( A \lor B \)
2. Option 2: \( A \land (A \implies B) \)
3. Option 3: \( B \implies (A \land B) \)
4. Option 4: \( A \land B \)

### Step 3: Create Truth Tables
We will create truth tables for each option to determine if they are tautologies.

#### Option 1: \( A \lor B \)
| A | B | \( A \lor B \) |
|-------|-------|----------------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

- This option is **not** a tautology since it evaluates to false when both A and B are false.

#### Option 2: \( A \land (A \implies B) \)
Using the implication equivalence \( A \implies B \equiv \neg A \lor B \):
- Rewrite: \( A \land (\neg A \lor B) \)

Now, we can create a truth table:
| A | B | \( A \implies B \) | \( A \land (A \implies B) \) |
|-------|-------|---------------------|-------------------------------|
| T | T | T | T |
| T | F | F | F |
| F | T | T | F |
| F | F | T | F |

- This option is **not** a tautology since it evaluates to false in three cases.

#### Option 3: \( B \implies (A \land B) \)
Using the implication equivalence:
- Rewrite: \( \neg B \lor (A \land B) \)

Truth table:
| A | B | \( A \land B \) | \( B \implies (A \land B) \) |
|-------|-------|------------------|-------------------------------|
| T | T | T | T |
| T | F | F | T |
| F | T | F | F |
| F | F | F | T |

- This option is **not** a tautology since it evaluates to false when A is false and B is true.

#### Option 4: \( A \land B \)
| A | B | \( A \land B \) |
|-------|-------|------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

- This option is **not** a tautology since it evaluates to false when either A or B is false.

### Conclusion
After analyzing all options, we find that none of the options are tautologies. However, if we had to select the one that is closest to being always true, we would select Option 2 based on the analysis.