Which of the following is a tautology

Options

A. `((p rarr q) wedge ~ q) rarr p wedgeq`
B. `((p rarr q) wedge ~ q) rarr p `
C. `((p rarr q) wedge ~ q) rarr q `
D. `((p rarr q) wedge ~ q) rarr ~q `

Answer

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

Let's denote the statements as follows:
1. Statement A: \( p \land q \)
2. Statement B: \( p \implies q \land \neg q \)
3. Statement C: \( p \implies q \)
4. Statement D: \( p \land \neg q \implies p \)

We will evaluate each statement step by step.

### Step 1: Construct the truth table for \( p \) and \( q \)

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

### Step 2: Evaluate \( p \implies q \)

The implication \( p \implies q \) is false only when \( p \) is true and \( q \) is false.

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

### Step 3: Evaluate \( \neg q \)

Negation of \( q \) is true when \( q \) is false.

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

### Step 4: Evaluate \( p \implies q \land \neg q \)

Now we will evaluate the conjunction \( p \implies q \land \neg q \).

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

### Step 5: Evaluate \( p \land q \)

Now we will evaluate the conjunction \( p \land q \).

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

### Step 6: Evaluate \( p \land \neg q \)

Now we will evaluate \( p \land \neg q \).

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

### Step 7: Evaluate \( p \land \neg q \implies p \)

Now we will evaluate the implication \( p \land \neg q \implies p \).

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

### Step 8: Analyze the results

Now we will analyze the results of the evaluations:

1. Statement A: \( p \land q \) is not a tautology (not always true).
2. Statement B: \( p \implies q \land \neg q \) is not a tautology (not always true).
3. Statement C: \( p \implies q \) is not a tautology (not always true).
4. Statement D: \( p \land \neg q \implies p \) is a tautology (always true).

### Conclusion

The tautology among the statements is **Statement D**: \( p \land \neg q \implies p \).

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