The number of values of `r in {p,q,~p,~q}` for which `(p^^q)implies (r vv q)^^((p ^^ r) implies q)` is a tautology,is :

Options

A. `1`
B. `2`
C. `4`
D. `3`

Answer

To solve the problem, we need to determine the number of values of \( r \) in the set \( \{p, q, \neg p, \neg q\} \) for which the expression

\[ (p \land q) \implies (r \lor q) \land ((p \land r) \implies q) \]

is a tautology. A tautology is a statement that is true in every possible interpretation.

### Step 1: Rewrite the Expression
We start by rewriting the expression using logical equivalences. The implication \( A \implies B \) can be rewritten as \( \neg A \lor B \).

Thus, we can rewrite the expression as:

\[ \neg (p \land q) \lor ((r \lor q) \land ((p \land r) \implies q)) \]

### Step 2: Simplify the Implication
Next, we simplify the second part of the expression:

\[ (p \land r) \implies q \text{ can be rewritten as } \neg (p \land r) \lor q \]

So, we have:

\[ \neg (p \land q) \lor ((r \lor q) \land (\neg (p \land r) \lor q)) \]

### Step 3: Analyze the Components
Now, we need to analyze the components of the expression. The expression will be a tautology if it evaluates to true for all combinations of truth values of \( p \) and \( q \).

1. **Case 1**: If \( p \) and \( q \) are both true, then \( (p \land q) \) is true, and we need \( (r \lor q) \land ((p \land r) \implies q) \) to be true.
2. **Case 2**: If \( p \) is true and \( q \) is false, then \( (p \land q) \) is false, and the expression is automatically true.
3. **Case 3**: If \( p \) is false and \( q \) is true, then again \( (p \land q) \) is false, and the expression is automatically true.
4. **Case 4**: If both \( p \) and \( q \) are false, then \( (p \land q) \) is false, and the expression is automatically true.

### Step 4: Determine Values of \( r \)
Now we need to check the values of \( r \) that make the expression true in case 1 (where both \( p \) and \( q \) are true):

- If \( r = p \): The expression becomes true.
- If \( r = q \): The expression becomes true.
- If \( r = \neg p \): The expression becomes false because \( (p \land r) \) becomes true and \( q \) is false.
- If \( r = \neg q \): The expression becomes false because \( (p \land r) \) becomes true and \( q \) is false.

Thus, the values of \( r \) that make the expression a tautology are \( p \) and \( q \).

### Conclusion
The number of values of \( r \) in the set \( \{p, q, \neg p, \neg q\} \) for which the expression is a tautology is **2**.