Consider the statement: If $x$ is a real number such that $x^2-5x+6=0$, then $x$ is a prime number. Which of the following is the contrapositive of this statement?

Solution

1. Let $P$ be the statement '$x^2-5x+6=0$' and $Q$ be the statement '$x$ is a prime number'.
2. The original implication is $P⟹Q$.
3. The contrapositive of $P⟹Q$ is defined as $\neg Q⟹\neg P$.
4. Negating $Q$ gives 'x is not a prime number', and negating $P$ gives '$x^2-5x+6\ne 0$'.
5. Combining these, the contrapositive is 'If $x$ is not a prime number, then $x^2-5x+6\ne 0$'.
6. Hence the answer is (A).