The contrapositive of the statement, 'If x is a prime number and x divides ab then x divides a or x divides b", can be symnolically represented using logical connectives, on appropriately defined statements p, q, r, s as

Options

A. `(~r vv ~s) rarr (~p ^^ ~q)`
B. `(r ^^ s) rarr (~p ^^ ~q)`
C. `(~r ^^ ~s) rarr (~p vv ~q)`
D. `(r vv s) rarr (~p vv ~q)`

Answer

To find the contrapositive of the statement "If x is a prime number and x divides ab, then x divides a or x divides b," we can break down the process step by step.

### Step-by-Step Solution:

1. **Define the Statements:**
- Let \( p \): "x is a prime number."
- Let \( q \): "x divides ab."
- Let \( r \): "x divides a."
- Let \( s \): "x divides b."

2. **Original Statement in Logical Form:**
- The original statement can be represented as:
\[ (p \land q) \implies (r \lor s) \]
- Here, \( \land \) represents "and," \( \lor \) represents "or," and \( \implies \) represents "implies."

3. **Negate the Conclusion and the Hypothesis:**
- The contrapositive of a statement \( A \implies B \) is \( \neg B \implies \neg A \).
- In our case, \( A \) is \( (p \land q) \) and \( B \) is \( (r \lor s) \).
- Therefore, we need to negate both:
- Negation of \( (r \lor s) \) is \( \neg r \land \neg s \) (using De Morgan's laws).
- Negation of \( (p \land q) \) is \( \neg p \lor \neg q \) (again using De Morgan's laws).

4. **Construct the Contrapositive:**
- Now we can write the contrapositive as:
\[ (\neg r \land \neg s) \implies (\neg p \lor \neg q) \]

5. **Final Representation:**
- Thus, the contrapositive of the original statement is:
\[ (\neg r \land \neg s) \implies (\neg p \lor \neg q) \]

### Conclusion:
The contrapositive of the statement "If x is a prime number and x divides ab, then x divides a or x divides b" can be symbolically represented as:
\[ (\neg r \land \neg s) \implies (\neg p \lor \neg q) \]