Question

Theorem: If x and y are positive real numbers then xy≥x or xy≥y

A proof by confradiction of the theorem start by assuming which fact?
a. There exists two real numbers, x and y, such that xyb. There exists two real numbers, x and y, such that xyc. There exists two real numbers, x and y, such that xy≥x and xy≥y
d. There exists two real numbers, x and y, guch that x y ≥x or xy≥ y

Answer 1

The correct assumption to start the proof by contradiction of the theorem is option d: There exists two real numbers, x and y, such that xy ≥ x or xy ≥ y.Option D is the correct answer.

Step-by-step explanation:

The correct assumption to start the proof by contradiction of the theorem is option d: There exists two real numbers, x and y, such that xy ≥ x or xy ≥ y.

To prove the theorem by contradiction, we assume the opposite of what we want to prove and show that it leads to a contradiction. In this case, we assume that there exists two real numbers x and y such that xy < x and xy < y. By making this assumption, we can proceed with the proof and eventually arrive at a contradiction, proving that the original theorem holds.

Starting the proof by contradiction with the assumption in option d, where there exist real numbers x and y such that xy ≥ x or xy ≥ y, provides a logical foundation. Contradicting this assumption leads to a proof of the original theorem. This method reinforces the theorem's validity, showcasing the power of proof by contradiction in establishing the truth of mathematical propositions through rigorous logical reasoning.