If p and q are true and r and s are false statements, then which of the following is true?

Options

A. `(q ^^ r) vv (~p ^^ s)`
B. `(~p to q)harr(r ^^ s)`
C. `(p to q) vv (r larr s)`
D. `(p ^^ ~r) ^^ (~ q vv s)`

Explanation

To solve the problem, we need to evaluate the truth values of the given statements based on the truth values of P, Q, R, and S.

Given:
- P = True
- Q = True
- R = False
- S = False

We will evaluate each option step by step.

### Step 1: Evaluate Option 1: Q and R
- Q is True.
- R is False.
- The conjunction (AND) of Q and R is:
\[ Q \land R = \text{True} \land \text{False} = \text{False} \]

### Step 2: Evaluate Option 2: Negation of P
- Negation of P (¬P) is:
\[ \neg P = \neg \text{True} = \text{False} \]
- Now we need to evaluate the conditional statement:
\[ \neg P \rightarrow Q \]
Since ¬P is False and Q is True, the conditional statement is:
\[ \text{False} \rightarrow \text{True} = \text{True} \]
- Now we evaluate the biconditional statement:
\[ R \leftrightarrow S \]
Since both R and S are False:
\[ \text{False} \leftrightarrow \text{False} = \text{True} \]
- Therefore, the overall evaluation for this option is:
\[ \text{True} \text{ (from the conditional)} \land \text{True} \text{ (from the biconditional)} = \text{True} \]

### Step 3: Evaluate Option 3: P conditional Q and R
- The conditional statement is:
\[ P \rightarrow Q \]
Since both P and Q are True:
\[ \text{True} \rightarrow \text{True} = \text{True} \]
- Now we evaluate:
\[ S \rightarrow R \]
Since both S and R are False:
\[ \text{False} \rightarrow \text{False} = \text{True} \]
- Therefore, the overall evaluation for this option is:
\[ \text{True} \land \text{True} = \text{True} \]

### Step 4: Evaluate Option 4: S conditional R
- The conditional statement is:
\[ S \rightarrow R \]
Since both S and R are False:
\[ \text{False} \rightarrow \text{False} = \text{True} \]

### Conclusion:
The only true statements from the evaluations are found in Option 2 and Option 3. However, since the question asks for which of the following is true, we find that Option 3 is indeed correct.

### Final Answer:
**Option 3 is true.**