Question

Create a truth table for the following statement: ~p^q

Answer 1

To create a truth table for the logical statement \( \sim p \land q \) (which reads as "not p and q"), we need to consider all possible truth values for the variables \( p \) and \( q \). The variables \( p \) and \( q \) can each be either true (T) or false (F).

Here's how you set up the truth table:

1. **Columns** for \( p \), \( q \), \( \sim p \), and \( \sim p \land q \).
2. **Rows** for all possible combinations of truth values for \( p \) and \( q \).

The combinations are:
- \( p = T, q = T \)
- \( p = T, q = F \)
- \( p = F, q = T \)
- \( p = F, q = F \)

Now, let's fill out the truth table:

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

### Explanation:
- **Column 1 (p)**: lists the truth value of \( p \).
- **Column 2 (q)**: lists the truth value of \( q \).
- **Column 3 (\( \sim p \))**: shows the negation of \( p \).
- **Column 4 (\( \sim p \land q \))**: shows the result of the conjunction (AND operation) between \( \sim p \) and \( q \).

From the table, we can see:
- When \( p \) is true, \( \sim p \) is false, resulting in \( \sim p \land q \) being false, regardless of the value of \( q \).
- When \( p \) is false and \( q \) is true, \( \sim p \) is true, which means \( \sim p \land q \) is true.
- When both \( p \) and \( q \) are false, \( \sim p \) is true but \( q \) is false, resulting in \( \sim p \land q \) being false.

Thus, the only scenario where \( \sim p \land q \) evaluates to true is when \( p \) is false and \( q \) is true.