If truth values of ` p vv q` is ture,then truth value of ` ~ p ^^ q` is

Options

A. false if p is true
B. true if p is true
C. false if q is true
D. true if q is true

Answer

To solve the problem, we need to analyze the given statement and find the truth value of `~p ∧ q` given that the truth value of `p ∨ q` is true.

### Step-by-Step Solution:

1. **Understanding the Given Statement**:
We are given that `p ∨ q` (p or q) is true. This means at least one of the propositions `p` or `q` must be true.

2. **Constructing the Truth Table for `p ∨ q`**:
Let's create a truth table for `p` and `q` to see when `p ∨ q` is true.

| p | q | p ∨ q |
|-------|-------|-------|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |

From the truth table, we see that `p ∨ q` is true in three cases:
- When both `p` and `q` are true (T, T)
- When `p` is true and `q` is false (T, F)
- When `p` is false and `q` is true (F, T)

3. **Finding the Truth Value of `~p ∧ q`**:
Now we need to evaluate `~p ∧ q` (not p and q) for the cases where `p ∨ q` is true.

- **Case 1**: `p = T`, `q = T`
- `~p = F`, `~p ∧ q = F ∧ T = F`
- **Case 2**: `p = T`, `q = F`
- `~p = F`, `~p ∧ q = F ∧ F = F`
- **Case 3**: `p = F`, `q = T`
- `~p = T`, `~p ∧ q = T ∧ T = T`

4. **Conclusion**:
From the evaluations, we see that `~p ∧ q` is true only in the third case where `p` is false and `q` is true. Therefore, the truth value of `~p ∧ q` can be either true or false depending on the values of `p` and `q`. However, since we are given that `p ∨ q` is true, the only case that satisfies this condition and makes `~p ∧ q` true is when `p` is false and `q` is true.

### Final Answer:
The truth value of `~p ∧ q` is **True** when `p` is false and `q` is true.