If p and q are true statements in logic, which of the following statement pattern is true?

Options

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

Answer

To determine which of the statement patterns is true given that both p and q are true statements, we can analyze each option step by step.

### Step-by-Step Solution:

1. **Identify the values of p and q**:
- Given that both p and q are true, we have:
- p = True
- q = True

2. **Evaluate each option**:

**Option 1**: \( p \land q \)
- Since both p and q are true, \( p \land q \) (p AND q) is also true.
- **Result**: True

**Option 2**: \( p \land \neg q \)
- Here, \( \neg q \) (negation of q) is false because q is true.
- Thus, \( p \land \neg q \) (p AND NOT q) is false.
- **Result**: False

**Option 3**: \( p \land \neg q \Rightarrow q \)
- We already established that \( \neg q \) is false.
- Therefore, \( p \land \neg q \) is false.
- In logic, a false antecedent (the part before the implication) makes the implication true regardless of the truth value of the consequent (the part after the implication).
- **Result**: True

**Option 4**: \( \neg p \land q \)
- Here, \( \neg p \) (negation of p) is false since p is true.
- Thus, \( \neg p \land q \) (NOT p AND q) is false.
- **Result**: False

3. **Conclusion**:
- The only true statement pattern among the options is from **Option 3**: \( p \land \neg q \Rightarrow q \).

### Final Answer:
The correct answer is **Option 3**: \( p \land \neg q \Rightarrow q \).