Question

In the theorem, for any two real numbers, x and y, if x and y are both rational, then x*y is also rational. Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

1) The fact that x and y are both rational is assumed, and the fact that x*y is rational is proven.
2) The fact that x and y are both rational is proven, and the fact that x*y is rational is assumed.
3) The fact that x and y are both rational is assumed and proven.
4) The fact that x and y are both rational and x*y is rational are both assumed.

Answer 1

In a proof by contrapositive for the theorem stating that the product of two rational numbers is rational, the assumption is that x and y are both rational, and it is proven that their product, x*y, is rational.

Step-by-step explanation:

The theorem in question states that for any two real numbers, x and y, if x and y are both rational, then x*y is also rational. When we are dealing with a proof by contrapositive, we assume the negation of the conclusion to prove the negation of the premise. Specifically, we would start with the assumption that x*y is not rational (irrational) and show that this leads to either x or y being irrational, thus proving that if x or y were irrational to begin with, then x*y would not be rational.

The correct answer to which facts are assumed and which are proven in a proof by contrapositive of the theorem is:

1) The fact that x and y are both rational is assumed, and the fact that x*y is rational is proven.