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: Reinterpreting the condition:
\((a,b)\,R\,(c,d)\iff ad=bc \iff \dfrac{a}{b}=\dfrac{c}{d}\) — i.e. the pair \((a,b)\) is related to \((c,d)\) exactly when they represent the same ratio.

Step 2: Reflexive and symmetric via ratios:
\(\dfrac{a}{b}=\dfrac{a}{b}\) trivially (reflexive); and \(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{c}{d}=\dfrac{a}{b}\) (symmetric).

Step 3: Transitive via ratios:
If \(\dfrac{a}{b}=\dfrac{c}{d}\) and \(\dfrac{c}{d}=\dfrac{e}{f}\), then \(\dfrac{a}{b}=\dfrac{e}{f}\), i.e. \(af=be\), giving \((a,b)\,R\,(e,f)\).

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