Which of the following is logically equivalent to ` ~( p to q) ` ?

Options

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

Answer

To find which of the following is logically equivalent to \( \sim (p \to q) \), we will use the truth table method.

### Step-by-Step Solution:

1. **Understanding Implication**:
The implication \( p \to q \) can be rewritten in terms of logical operators:
\[ p \to q \equiv \sim p \lor q \]
This means that \( p \to q \) is false only when \( p \) is true and \( q \) is false.

2. **Negating the Implication**:
We need to find \( \sim (p \to q) \):
\[ \sim (p \to q) \equiv \sim (\sim p \lor q) \]
By applying De Morgan's laws, this can be rewritten as:
\[ \sim (\sim p \lor q) \equiv p \land \sim q \]

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

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

From the table, we see that \( \sim (p \to q) \) is true only when \( p \) is true and \( q \) is false.

4. **Identifying Logical Equivalence**:
We have established that \( \sim (p \to q) \equiv p \land \sim q \). Thus, we need to check which option matches this expression.

5. **Conclusion**:
The expression \( p \land \sim q \) is logically equivalent to \( \sim (p \to q) \).

### Final Answer:
The expression that is logically equivalent to \( \sim (p \to q) \) is \( p \land \sim q \).

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