In the truth table for the statements ` ( p to q) harr(~p vvq)`, the last column has the truth value in the following order

Options

A. (a) TTTT
B. (b) FTFT
C. (c) TTFF
D. (d) FFFF

Explanation

To solve the question regarding the truth table for the statements \( (p \to q) \leftrightarrow (\neg p \lor q) \), we will follow these steps:

### Step 1: Create the Truth Table
We need to create a truth table that includes the columns for \( p \), \( q \), \( \neg p \), \( p \to q \), \( \neg p \lor q \), and finally \( (p \to q) \leftrightarrow (\neg p \lor q) \).

| \( p \) | \( q \) | \( \neg p \) | \( p \to q \) | \( \neg p \lor q \) | \( (p \to q) \leftrightarrow (\neg p \lor q) \) |
|---------|---------|---------------|----------------|----------------------|--------------------------------------------------|
| T | T | F | T | T | ? |
| T | F | F | F | F | ? |
| F | T | T | T | T | ? |
| F | F | T | T | T | ? |

### Step 2: Fill in the Negation of \( p \)
Negation of \( p \) is simply the opposite of \( p \):
- If \( p \) is True (T), then \( \neg p \) is False (F).
- If \( p \) is False (F), then \( \neg p \) is True (T).

So, we fill in the column for \( \neg p \):
| \( p \) | \( q \) | \( \neg p \) |
|---------|---------|---------------|
| T | T | F |
| T | F | F |
| F | T | T |
| F | F | T |

### Step 3: Calculate \( p \to q \)
The implication \( p \to q \) is only false when \( p \) is true and \( q \) is false. Otherwise, it is true:
- For \( (T, T) \): True
- For \( (T, F) \): False
- For \( (F, T) \): True
- For \( (F, F) \): True

So, we fill in the column for \( p \to q \):
| \( p \) | \( q \) | \( \neg p \) | \( p \to q \) |
|---------|---------|---------------|----------------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

### Step 4: Calculate \( \neg p \lor q \)
The disjunction \( \neg p \lor q \) is true if at least one of \( \neg p \) or \( q \) is true:
- For \( (T, T) \): True
- For \( (T, F) \): False
- For \( (F, T) \): True
- For \( (F, F) \): True

So, we fill in the column for \( \neg p \lor q \):
| \( p \) | \( q \) | \( \neg p \) | \( p \to q \) | \( \neg p \lor q \) |
|---------|---------|---------------|----------------|----------------------|
| T | T | F | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | F | T | T | T |

### Step 5: Calculate \( (p \to q) \leftrightarrow (\neg p \lor q) \)
The biconditional \( (p \to q) \leftrightarrow (\neg p \lor q) \) is true if both sides have the same truth value:
- For \( (T, T) \): True
- For \( (F, F) \): True
- For \( (T, F) \): False
- For \( (T, T) \): True

So, we fill in the last column:
| \( p \) | \( q \) | \( \neg p \) | \( p \to q \) | \( \neg p \lor q \) | \( (p \to q) \leftrightarrow (\neg p \lor q) \) |
|---------|---------|---------------|----------------|----------------------|--------------------------------------------------|
| T | T | F | T | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |

### Final Truth Values
The last column has the truth values in the order: T, T, T, T.

### Conclusion
The correct answer is that the last column has the truth value in the order: **T, T, T, T**.