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

Question

Responses

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

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

Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.

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

Answer 1

The correct response is:

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

However, it's important to clarify that this statement is actually incorrect in terms of the sum of a rational and an irrational number; the correct statement should be:

"The sum or difference of a rational number and an irrational number is an irrational number."

In defining the closure property, it means that performing a certain operation (like addition or subtraction) on members of a set results in a member of the same set. For rational numbers, their sum or difference remains a rational number. But when you mix a rational number with an irrational number, the result is outside of the rational set, leading to an irrational number.