Question

N is the set of positive integers and ~ be a relation on $N\times N$ defined (a, b) ~ (c, d) iff ad = bc.Is it equivalence relation.

Answer 1

Consider any $(a,b),(c,d),(e,f)\in N\times N$.

(i) ab = ba for all $a,b\in N$ (commutative law of multiplication in N)

$\Rightarrow (a,b)R(a,b)\Rightarrow R$ is reflexive.

(ii) Let $(a,b)R(c,d)\Rightarrow ad=bc\Rightarrow bc=ad$

$\Rightarrow cb=da\Rightarrow (c,d)R(a,b)$.

Thus, $(a,b)R(c,d)\Rightarrow (c,d)R(a,b)\Rightarrow R$ is symmetric.

(iii) Let (a, b) R (c, d) and (c, d) R (e, f )

$\Rightarrow ad=bcandcf=de\Rightarrow adcf=bcde$

$\Rightarrow af=be\Rightarrow (a,b)R(e,f)$.

Thus, (a, b) R (c, d ) and $(c,d)R(e,f)\Rightarrow (a,b)R(e,f)\Rightarrow R$ is transitive.

Therefore, the relation R is reflexive, symmetric and transitive, and hence it is an equivalence relation