A relation \( R \) is defined in \( \mathbb{N} \times \mathbb{N} \) as follows:
\[
(a, b) \, R \, (c, d) { if and only if } ad = bc.
\]
Prove that \( R \) is an equivalence relation.
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Solution and Explanation
Step 1: Reflexivity: For reflexivity, we need \( (a, b) \, R \, (a, b) \), i.e., \( ad = bc \). Clearly, \( a \cdot b = b \cdot a \), so \( R \) is reflexive.
Step 2: Symmetry: For symmetry, we need that if \( (a, b) \, R \, (c, d) \), i.e., \( ad = bc \), then \( (c, d) \, R \, (a, b) \). Since \( ad = bc \), we have \( bc = ad \), thus symmetry holds.
Step 3: Transitivity: For transitivity, if \( (a, b) \, R \, (c, d) \) and \( (c, d) \, R \, (e, f) \), then we need \( (a, b) \, R \, (e, f) \). From \( ad = bc \) and \( cf = de \), we get \( ad \cdot cf = bc \cdot de \), confirming that transitivity holds.
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