Which of the following statements is tautology ?

Options

A. `(~q ^^ p)^^ q`
B. `(~q ^^ p)^^(p ^^ ~q)`
C. `(~q ^^ p)vv(p vv ~p)`
D. `(p ^^ q)^^(~(p ^^ q))`

Explanation

To determine which of the given statements is a tautology, we need to analyze each statement using truth tables. A tautology is a statement that is true in every possible interpretation.

Let's denote the statements as follows:
1. \( T_1: \neg q \land p \)
2. \( T_2: p \land \neg q \)
3. \( T_3: \neg q \land p \lor (p \lor \neg p) \)
4. \( T_4: p \land q \)

### Step 1: Identify the variables and their possible values
We have two variables \( p \) and \( q \). Each variable can be either true (T) or false (F). The possible combinations of truth values for \( p \) and \( q \) are as follows:

| p | q |
|-------|-------|
| T | T |
| T | F |
| F | T |
| F | F |

### Step 2: Construct truth tables for each statement

#### Truth Table for \( T_1: \neg q \land p \)

| p | q | \neg q | \( \neg q \land p \) |
|-------|-------|--------|-----------------------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |

#### Truth Table for \( T_2: p \land \neg q \)

| p | q | \neg q | \( p \land \neg q \) |
|-------|-------|--------|-----------------------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |

#### Truth Table for \( T_3: \neg q \land p \lor (p \lor \neg p) \)

Since \( p \lor \neg p \) is always true (a tautology), we can simplify our analysis:

| p | q | \neg q | \( \neg q \land p \) | \( p \lor \neg p \) | \( T_3 \) |
|-------|-------|--------|-----------------------|----------------------|-----------|
| T | T | F | F | T | T |
| T | F | T | T | T | T |
| F | T | F | F | T | T |
| F | F | T | F | T | T |

#### Truth Table for \( T_4: p \land q \)

| p | q | \( p \land q \) |
|-------|-------|------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

### Step 3: Analyze the results
- \( T_1 \): F, T, F, F (not a tautology)
- \( T_2 \): F, T, F, F (not a tautology)
- \( T_3 \): T, T, T, T (this is a tautology)
- \( T_4 \): T, F, F, F (not a tautology)

### Conclusion
The statement that is a tautology is \( T_3: \neg q \land p \lor (p \lor \neg p) \).

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