Question

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.

Hint:

To prove a relation is an equivalence relation, verify that it is reflexive, symmetric, and transitive.

Answer 1

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.