Question

A relation \(R\) is defined in \(N\times N\) as follows: \((a,b)\,R\,(c,d)\) if and only if \(ad=bc\). Prove that \(R\) is an equivalence relation.

Hint:

Check reflexive, symmetric, transitive directly from ad = bc.

Answer 1

Step 1: Reflexivity:
For any \((a,b)\in N\times N\): \(ab=ba\) is always true, so \((a,b)\,R\,(a,b)\). Hence \(R\) is reflexive.

Step 2: Symmetry:
Suppose \((a,b)\,R\,(c,d)\), i.e. \(ad=bc\). Then \(cb=da\), i.e. \(cb=da\) which is exactly the condition \((c,d)\,R\,(a,b)\). Hence \(R\) is symmetric.

Step 3: Transitivity:
Suppose \((a,b)\,R\,(c,d)\) and \((c,d)\,R\,(e,f)\), i.e. \(ad=bc\) and \(cf=de\). Multiply: \(ad\cdot cf=bc\cdot de \Rightarrow adcf=bcde\). Cancel the common factor \(cd\) (nonzero, natural numbers): \(af=be\), which is \((a,b)\,R\,(e,f)\). Hence \(R\) is transitive.

Final Answer:
Since \(R\) is reflexive, symmetric and transitive, \(R\) is an equivalence relation.\[ \boxed{R \text{ is an equivalence relation}} \]