`p implies q` can also be written as-

Options

A. `p implies ~q`
B. `~ p vv q`
C. `~ p implies ~q`
D. None of these

Answer

To solve the question "p implies q can also be written as", we will use the truth table method to analyze the logical implications.

### Step-by-Step Solution:

1. **Understanding Implication**:
The implication \( p \implies q \) can be defined in terms of truth values:
- It is **false** only when \( p \) is true and \( q \) is false.
- In all other cases (when \( p \) is false or \( q \) is true), it is **true**.

2. **Constructing the Truth Table**:
We will create a truth table for \( p \), \( q \), and \( p \implies q \).

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

3. **Finding Equivalent Expressions**:
We need to find an expression that has the same truth values as \( p \implies q \).

- **Negation of p**: \( \neg p \) will be true when \( p \) is false.
- **Negation of q**: \( \neg q \) will be true when \( q \) is false.

We will check the following options:
- \( \neg p \)
- \( \neg p \lor q \)
- \( \neg p \implies \neg q \)
- None of these

4. **Calculating \( \neg p \) and \( \neg q \)**:
We will also create a truth table for \( \neg p \) and \( \neg q \).

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

5. **Calculating \( \neg p \lor q \)**:
Now we will calculate \( \neg p \lor q \).

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

6. **Calculating \( \neg p \implies \neg q \)**:
Finally, we will calculate \( \neg p \implies \neg q \).

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

7. **Comparing Truth Values**:
Now we compare the truth values of \( p \implies q \) with \( \neg p \lor q \) and \( \neg p \implies \neg q \).

- \( p \implies q \): T, F, T, T
- \( \neg p \lor q \): T, F, T, T
- \( \neg p \implies \neg q \): T, T, F, T

From the comparison, we see that \( p \implies q \) is equivalent to \( \neg p \lor q \).

### Conclusion:
Thus, the expression \( p \implies q \) can also be written as \( \neg p \lor q \).