The question asks about the theorem:

"For any two real numbers, $x$ and $y$, if $x$ and $y$ are both rational, then $x + y$ is also rational."

The task is to identify which assumptions and proven facts match this theorem.

Let's break down the options:

• Option A:

  • Assumed: $x$ is rational and $y$ is rational.
  • Proven: $x + y$ is rational.
  • This matches the theorem exactly, where we assume that both $x$ and $y$ are rational, and we are proving that their sum is rational.

• Option B:

  • Assumed: $x + y$ is rational.
  • Proven: $x$ is rational or $y$ is rational.
  • This is not correct because the theorem is proving something about the sum of two rational numbers, not making assumptions about the sum first.

• Option C:

  • Assumed: $x$ is rational or $y$ is rational.
  • Proven: $x + y$ is rational.
  • This is incorrect because the theorem requires both $x$ and $y$ to be rational to guarantee their sum is rational.

• Option D:

  • Assumed: $x + y$ is irrational.
  • Proven: $x$ is irrational and $y$ is irrational.
  • This is not related to the theorem at all, as the theorem deals with sums of rational numbers, not irrational numbers.

Correct Answer: A

Would you like more details on the reasoning? Here are some related questions:

1. What defines a rational number mathematically?
2. Can the sum of a rational number and an irrational number ever be rational?
3. What is the difference between a direct and an indirect proof?
4. How can you prove that the sum of two rational numbers is always rational?
5. Why is option B an incorrect assumption about the sum of rational numbers?

Tip: When proving properties of numbers, always check the definitions (like rational or irrational) and see if they hold under operations such as addition or multiplication.