a. When asked to prove that the difference of any irrational number and any rational number is irrational, a student began, “Suppose not. That is, suppose the difference of any irrational number and any rational number is rational.” What is wrong with beginning the proof in this way? (Hint: Review the answer to exercise.)
b. Prove that the difference of any irrational number and any rational number is irrational.
Exercise
The proposed negation is not correct. Consider the given statement: “The sum of any two irrational numbers is irrational.” For this to be false means that it is possible to find at least one pair of irrational numbers whose sum is rational. On the other hand, the negation proposed in the exercise (“The sum of any two irrational numbers is rational”) means that given any two irrational numbers, their sum 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 are at least two irrational numbers), but the negation can be true without having this statement be true.
Correct negation: There are at least two irrational numbers whose sum is rational.
Or: The sum of some two irrational numbers is rational.
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