Let p and q be two statements. Amongst the following , the statement that is equivalent to ` p to q` is

Options

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

Explanation

To determine which statement is equivalent to "p to q" (denoted as \( p \implies q \)), we will create a truth table and analyze the options.

### Step 1: Create the Truth Table for \( p \implies q \)

1. **Identify the possible values for \( p \) and \( q \)**:
- Each statement can either be True (T) or False (F).
- Since there are two statements, there are \( 2^2 = 4 \) combinations.

2. **List the combinations**:
- \( p \) can be T or F.
- \( q \) can also be T or F.
- The combinations are:
- \( p = T, q = T \)
- \( p = T, q = F \)
- \( p = F, q = T \)
- \( p = F, q = F \)

3. **Determine the truth values for \( p \implies q \)**:
- The implication \( p \implies q \) is only false when \( p \) is true and \( q \) is false. In all other cases, it is true.
- The truth values for \( p \implies q \) are:
- \( T \implies T \) = T
- \( T \implies F \) = F
- \( F \implies T \) = T
- \( F \implies F \) = T

4. **Construct the truth table**:

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

### Step 2: Analyze the Options

Now we will analyze the options provided to find an equivalent statement to \( p \implies q \).

1. **Negation of \( q \)**:
- Truth table for \( \neg q \):
- \( q = T \) → \( \neg q = F \)
- \( q = F \) → \( \neg q = T \)
- This does not match \( p \implies q \).

2. **Negation of \( p \) or \( q \)**:
- Truth table for \( \neg p \lor q \):
- \( p = T, q = T \) → \( \neg p = F \), \( \neg p \lor q = T \)
- \( p = T, q = F \) → \( \neg p = F \), \( \neg p \lor q = F \)
- \( p = F, q = T \) → \( \neg p = T \), \( \neg p \lor q = T \)
- \( p = F, q = F \) → \( \neg p = T \), \( \neg p \lor q = T \)
- This does not match \( p \implies q \).

3. **Negation of \( p \) and \( q \)**:
- Truth table for \( \neg p \land q \):
- \( p = T, q = T \) → \( \neg p = F \), \( \neg p \land q = F \)
- \( p = T, q = F \) → \( \neg p = F \), \( \neg p \land q = F \)
- \( p = F, q = T \) → \( \neg p = T \), \( \neg p \land q = T \)
- \( p = F, q = F \) → \( \neg p = T \), \( \neg p \land q = F \)
- This does not match \( p \implies q \).

4. **Disjunction of \( \neg p \) and \( q \)**:
- Truth table for \( \neg p \lor q \):
- \( p = T, q = T \) → \( \neg p = F \), \( \neg p \lor q = T \)
- \( p = T, q = F \) → \( \neg p = F \), \( \neg p \lor q = F \)
- \( p = F, q = T \) → \( \neg p = T \), \( \neg p \lor q = T \)
- \( p = F, q = F \) → \( \neg p = T \), \( \neg p \lor q = T \)
- This matches \( p \implies q \).

### Conclusion

The statement that is equivalent to \( p \implies q \) is \( \neg p \lor q \).