Question

Theorem: There is no smallest positive rational number. A proof by contradiction of the theorem starts by assuming which fact?

1) There is a smallest positive rational number
2) There is a largest positive rational number
3) There is an infinite number of positive rational numbers
4) There is a negative rational number

Answer 1

To prove there's no smallest positive rational number, we'd assume that there is a smallest one and then demonstrate a contradiction by finding a smaller number, thus proving the original theorem by contradiction.

Step-by-step explanation:

To prove the theorem that there is no smallest positive rational number, we would start by assuming the opposite of the theorem. A proof by contradiction takes the negation of what you are trying to prove and then shows that such an assumption leads to a contradiction. Therefore, the proof would begin by assuming that there is a smallest positive rational number.

This assumption is then shown to be flawed by considering a smaller number. For instance, if we assumed that r is the smallest positive rational number, we could divide r by 2 to get a smaller positive rational number, which is a contradiction to our initial assumption that r was the smallest. This logical contradiction confirms that our assumption is false, thus proving the original theorem that there is no smallest positive rational number.