Which one of the following Boolean expression is a tautology ?

Options

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

Answer

To determine which Boolean expression is a tautology, we need to evaluate 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 variables as follows:
- \( p \): a Boolean variable
- \( q \): another Boolean variable

We will evaluate the following expressions:
1. \( p \land q \lor p \implies q \) (Option A)
2. \( p \lor q \land p \implies q \) (Option B)
3. \( p \land q \land p \implies q \) (Option C)
4. \( p \land q \implies p \implies q \) (Option D)

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

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

### Step 2: Evaluate each expression

#### Option A: \( p \land q \lor p \implies q \)

1. Calculate \( p \land q \):
- T and T = T
- T and F = F
- F and T = F
- F and F = F

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

2. Calculate \( p \implies q \):
- T implies T = T
- T implies F = F
- F implies T = T
- F implies F = T

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

3. Combine \( p \land q \) and \( p \implies q \):
- T or T = T
- F or F = F
- F or T = T
- F or T = T

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

This expression is **not a tautology** because it has a false value.

#### Option B: \( p \lor q \land p \implies q \)

1. Calculate \( p \lor q \):
- T or T = T
- T or F = T
- F or T = T
- F or F = F

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

2. Combine with \( p \implies q \) (calculated earlier).

3. Evaluate \( (p \lor q) \land (p \implies q) \):
- T and T = T
- T and F = F
- T and T = T
- F and T = F

This expression is also **not a tautology**.

#### Option C: \( p \land q \land p \implies q \)

1. Calculate \( p \land q \) (already calculated).
2. Combine with \( p \implies q \) (already calculated).

This expression is also **not a tautology**.

#### Option D: \( p \land q \implies p \implies q \)

1. Calculate \( p \land q \) (already calculated).
2. Combine with \( p \implies q \) (already calculated).

This expression is **a tautology** because all truth values are true.

### Conclusion

The Boolean expression that is a tautology is **Option D: \( p \land q \implies p \implies q \)**.