Let be the set of all ordered pairs of natural numbers and R be the relation on the set A defined by (a, b) R (c, d) iff ad = bc. Show that

Let $A= N\times N$ be the set of all ordered pairs of natural numbers and R be
the relation on the set A defined by (a, b) R (c, d) iff ad = bc. Show that R
is an equivalence relation.

Answers (1)

(i) Reflexive: $\left ( \left ( a,b \right ) ,\left ( a,b \right )\right )\epsilon R$
ab = ba
$\therefore \left ( \left ( a,b \right ),\left ( a,b \right ) \right )\epsilon R\Rightarrow R$ is reflexive
(ii) Symmetric: $(a,b),\left ( c,d \right )\epsilon R\Rightarrow \left ( \left ( c,d \right ),\left ( a,b \right ) \right )\epsilon R$
let $\left ( \left ( a,b \right )\left ( c,d \right ) \right )\epsilon R$
Let $\left ( \left ( a,b \right )\left ( c,d \right ) \right )\epsilon R$
Toprove $\left ( \left ( c,d \right )\left ( a,b \right ) \right )\epsilon R$ ie cb = da
Proof : $ad= bc\Rightarrow bc= ad$
ie $cb= da$
$\therefore \left ( \left ( c,d \right ),\left ( a,b \right ) \right )\epsilon R\: \Rightarrow$ R is symmetric.
(iii) Transitive: $\left ( \left ( a,b \right ),\left ( c,d \right )\epsilon R \right ),\left ( \left ( c,d \right ),\left ( e,f \right ) \right )\epsilon R$
      $\Rightarrow \left ( \left ( a,b \right ) ,\left ( e,f \right )\right )\epsilon R$
Let $\left ( \left ( a,b \right ),\left ( c,d \right ) \right )\epsilon R$ & $\left ( \left ( c,d \right ),\left ( e,f \right ) \right )\epsilon R$
Toprove $\left ( \left ( a,b \right ),\left ( e,f \right ) \right )\epsilon R \: \: ie\: af= be$
Proof : $ad= bc---(1)\: \: cf= de---(2)$
           muliplying eq (1) & (2)
    $a\not{d}\not{c}f= b\not{c}\not{d}e$
$\therefore af= be$
Hence $\left ( \left ( a,b \right ),\left ( e,f \right ) \right )\epsilon R\Rightarrow$ R is Transitive
Since the relation is reflexive, symmetric & transitive
Hence R is an equivalence relation.