`p wedge ~q` is equivalent to

Options

A. `~(p to q)`
B. `~(q to p)`
C. `(q to p)`
D. `(p to q)`

Answer

To determine the equivalence of the expression \( p \wedge \neg q \), we can use a truth table. Here's a step-by-step solution:

### Step 1: Create a Truth Table
We will create a truth table for the variables \( p \) and \( q \).

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

### Step 2: Calculate \( \neg q \)
Next, we will calculate \( \neg q \) (the negation of \( q \)).

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

### Step 3: Calculate \( p \wedge \neg q \)
Now, we will calculate \( p \wedge \neg q \) (the conjunction of \( p \) and \( \neg q \)).

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

### Step 4: Analyze the Results
From the truth table, we see that \( p \wedge \neg q \) is true only when \( p \) is true and \( q \) is false. The result is as follows:

- \( p \wedge \neg q \) is true for the row where \( p = T \) and \( q = F \) (2nd row).
- In all other cases, it is false.

### Step 5: Check the Options
Now, we need to check which of the given options is equivalent to \( p \wedge \neg q \):

1. **Option 1: \( \neg p \implies q \)**
Let's calculate \( \neg p \implies q \):
- \( \neg p \) is true when \( p \) is false.
- The implication \( \neg p \implies q \) is false only when \( \neg p \) is true and \( q \) is false.

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

This matches \( p \wedge \neg q \) (only false when \( p = F \) and \( q = F \)).

2. **Option 2: \( p \implies q \)**
This is false when \( p = T \) and \( q = F\), which does not match.

3. **Option 3: \( q \implies p \)**
This is false when \( q = T \) and \( p = F\), which does not match.

4. **Option 4: \( \neg p \wedge q \)**
This is true when \( p = F \) and \( q = T\), which does not match.

### Final Answer
The correct equivalence for \( p \wedge \neg q \) is **Option 1: \( \neg p \implies q \)**.

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