The contra positive of ` ( ~ p ^^ q) to ~ r` is

Options

A. ` ( p ^^q)to r`
B. ` ( p vv q) to r`
C. ` r to ( p vv ~ q)`
D. none of these

Answer

To find the contrapositive of the expression \( ( \neg p \land q) \to \neg r \), we will follow a systematic approach. The contrapositive of an implication \( A \to B \) is given by \( \neg B \to \neg A \).

### Step-by-Step Solution:

1. **Identify the components of the implication**:
- Here, \( A \) is \( \neg p \land q \) and \( B \) is \( \neg r \).

2. **Negate \( B \)**:
- We need to find \( \neg B \), which is \( \neg(\neg r) \).
- This simplifies to \( r \).

3. **Negate \( A \)**:
- Next, we need to find \( \neg A \), which is \( \neg(\neg p \land q) \).
- Using De Morgan's laws, this can be rewritten as \( \neg(\neg p) \lor \neg q \).
- This simplifies to \( p \lor \neg q \).

4. **Form the contrapositive**:
- Now, we can write the contrapositive as \( r \to (p \lor \neg q) \).

5. **Final expression**:
- Therefore, the contrapositive of \( ( \neg p \land q) \to \neg r \) is \( r \to (p \lor \neg q) \).

### Final Answer:
The contrapositive of \( ( \neg p \land q) \to \neg r \) is \( r \to (p \lor \neg q) \).

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