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LetR be a relation defined N×Ndefined by R={(a,b)R(c,d) if and ony if ad=b

LetR be a relation defined N×Ndefined by R={(a,b)R(c,d) if and ony if ad=bcand a,b,c,d€N} prove that R is an equivalence relation

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A)

∀(a,b)∈N×N∗,ab=ab

So (a,b)R(a,b) and R is reflexive.

 

Assume now that (a,b)R(c,d) i.e ad=bc.

Multiplication is commutative so

cb=da and

(c,d)R(a,b).

 The relation is symmetric

 

Now take (a,b)R(c,d) and (c,d)R(e,f) ;

this means ad=bc and cf=de.

 Multiply the first equality by f≠0 to get afd=bcf

 and the second by b≠0 to get bcf=bed.

 So we have afd=bed and

d≠0 we have

af=be

i.e (a,b)R(e,f) and the relation is transitive

 

It is therefore an equivalence relation

Now suppose (a,b)≃(c,d) and (c,d)≃(e,f)

Then ad=bc and cf=de

Thus

(ad)(cf)=(bc)(de)

and by cancelling from both sides

 

af=be

Accordingly (a,b)≃(e,f) and R is transitive.

R is an equivalence relation

 

 

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