Answer
The Proposed Negation is wrong.
Correct Negation: There are at least one irrational number and one rational number whose product is rational.
Work Step by Step
The proposed negation is not correct.
Consider the given statement: “The product of any irrational number and any rational number is irrational.” For this to be false means that it is possible to find at least one irrational number and one rational number whose product is rational.
On the other hand, the negation proposed in the exercise (“The product of any irrational number and any rational number is rational.”) means that given an irrational number and a rational number, their product is rational.
This is a much stronger statement than the actual negation: The truth of this statement implies the truth of the negation (assuming that there is at least an irrational number and a rational number), but the negation can be true without having this statement be true.
Correct negation: There are at least one irrational number and one rational number whose product is rational.
