To prove that the sum of a rational and an irrational number is irrational using the Closure Property, we can analyze the properties of rational and irrational numbers.
- The sum or difference of two rational numbers is a rational number. (True, but not directly relevant to this question.)
- The sum or difference of a rational and irrational number is a rational number. (False; this is the statement we want to prove is incorrect.)
- Irrational numbers can be added or subtracted. (True, but does not help in proving the specific statement.)
- The sum of an irrational and rational number can’t be predicted. (Vague, but does imply uncertainty regarding the outcome.)
The most relevant to the question is to note that:
- If \( r \) is a rational number and \( i \) is an irrational number, the assumption that \( r + i \) is rational leads to a contradiction. Specifically, if \( r + i \) were rational, then \( i \) could be expressed as the difference of two rational numbers (\( i = (r + i) - r \)), which contradicts the definition of an irrational number.
Therefore, the statement that “the sum or difference of a rational and irrational number is a rational number” is incorrect; it is the opposite that is true: the sum of a rational number and an irrational number is always irrational.
In this case, none of the provided responses correctly capture the necessary conclusion, which can be summarized as follows: "The sum of a rational number and an irrational number is irrational."