How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)

Question

Responses

A) The sum of an irrational and rational number can’t be predicted.

B) Irrational numbers can be added or subtracted.

C) The sum or difference of a rational and irrational number is a rational number.

D) The sum or difference of two rational numbers is a rational number.

Answer 1

The correct response is:

C) The sum or difference of a rational and irrational number is a rational number.

However, this statement is actually incorrect; the correct statement is that the sum or difference of a rational number and an irrational number is always irrational. Therefore, the closure property demonstrates that the sum of a rational number and an irrational number cannot be contained in the set of rational numbers, confirming that it must be irrational.

If you're looking for a better formulation, the logical reasoning would indicate that because the rational numbers are closed under addition and the result must always belong to a different set (the set of irrational numbers) when combined with an irrational number, the sum of a rational and an irrational number is indeed irrational.

In conclusion, the best choice based on the context of rationality would acknowledge that the sum of a rational and an irrational number is not rational, which is best captured in the negation of option C.