The paragraph I circled in blue can you explain that to me in more detail? Why c

Question

The paragraph I circled in blue can you explain that to me in more detail? Why can't we use the proposed negation above?

The statement is shown below, The product of any irrational number and any rational number is irrational" The proposed negation is The product of any irrational number and any rational number is rational" The objective is to determine if the proposed negation is correct The proposed negation is not correct because for the statement The product of any irrational number and any rational number is irrational to be false there must be at least one pair of irrational and rational numbers whose product is rational. The proposed negation "The product of any irrational number and any rational number is rational means that the product of any two given irrational and rational numbers is rational The truth of this statement implies the truth of the negation, but the negation can be rue without having its statement true. Therefore, the correct negation is There are at least a rational and an irrational number whose product is rational"

Explanation / Answer

Statement: "The product of any irrational number and any rational number is irrational".

Proposed Negation: "The product of any irrational number and any rational number is rational"

So, for the statement to be true, the product of ALL irrational and rational numbers has to be irrational.

Similarly, for the proposed negation to be true, the product of ALL irrational and rational numbers has to be rational.

But, to prove that the statement is false, we have to prove that the product of ANY irrational and rational number is rational.

Hence the circled statement that the negation can be true wihout having its statement true.

So the correct negation statement should be :The product of at least one pair of irrational number and rational number is rational"

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