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Let R be a relation on the set A of orde...

Let R be a relation on the set A of ordered pairs of positive integers defined by `(x , y) R (u , v)`if and only if `x v = y u`. Show that R is an equivalence relation.

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To show that the relation \( R \) defined on the set \( A \) of ordered pairs of positive integers is an equivalence relation, we need to demonstrate that it satisfies three properties: reflexivity, symmetry, and transitivity. ### Step 1: Check Reflexivity A relation is reflexive if for every element \( (x, y) \) in \( A \), it holds that \( (x, y) R (x, y) \). - For \( (x, y) R (x, y) \), we need to check if \( x \cdot y = y \cdot x \). - Since multiplication is commutative, \( x \cdot y = y \cdot x \) is always true. ...
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