To prove that the sum of a rational and an irrational number is irrational, we can refer to the Closure Property. The correct response relating to this proof is:
The sum or difference of a rational and irrational number is a rational number.
This statement is incorrect regarding our intent, as it should actually state that the sum or difference of a rational and irrational number is an irrational number.
Here's the reasoning:
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Definition of rational and irrational numbers: A rational number can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed as such a fraction.
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Assuming the opposite: If you assume that the sum of a rational number \( r \) and an irrational number \( i \) is rational, you could express it as \( r + i = x \) for some rational \( x \). Rearranging gives \( i = x - r \).
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Conclusion: Since both \( x \) and \( r \) are rational (by assumption), their difference \( x - r \) must also be rational. This implies that \( i \) is rational, which contradicts the assumption that \( i \) is irrational.
Thus, the sum of a rational number and an irrational number must indeed be irrational. The correct phrasing for the property should reflect that the sum or difference of a rational and irrational number is irrational, rather than rational.
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