It seems like you're discussing properties related to the addition of rational numbers and possibly their proofs. However, the phrasing is a bit unclear.
From what I can gather, you're asking to identify the property that relates to the combination of two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\).
Here are the three properties you've mentioned and what they typically signify:
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Associative Property: This property states that the way numbers are grouped does not affect their sum or product. For example, \((x + y) + z = x + (y + z)\).
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Closure Property: This property indicates that performing an operation (like addition or multiplication) on members of a set will always produce another member of the same set. For rational numbers, it states that the sum or product of two rational numbers is also a rational number.
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Commutative Property: This property asserts that the order in which two numbers are added or multiplied does not change the result. For example, \(x + y = y + x\).
If you are specifically looking for the property that states that the sum of \(a/b\) and \(c/d\) results in a rational number, the correct answer would be Closure Property (Option 2). This property guarantees that adding two rational numbers produces another rational number, confirming that the set of rational numbers is closed under addition.
Let me know if you need further clarification!
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