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A-relation-R-defined-by-x-y-R-u-v-v-2-y-2-u-2-x-2-show-that-R-is-an-equivalent-Relation-

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Question Number 67688 by Rio Michael last updated on 30/Aug/19
A relation R defined by _((x,y)) R_((u,v)) ⇔ v^2 −y^2 = u^2 −x^2 show that R is an equivalent Relation.
ArelationRdefinedby(x,y)R(u,v)⇔v2−y2=u2−x2showthatRisanequivalentRelation.
Commented by Prithwish sen last updated on 30/Aug/19
R_((x,y)) ⇒ y^2 −y^2 = x^2 −x^2 ⇒symmetric v^2 −y^2 = u^2 −x^2 ⇔y^2 −v^2 =x^2 −u^2 ⇒_((x,y)) R_((u,v)) ⇒_((u,v)) R_((x,y)) i.e reflexive v^2 −y^2 = u^2 −x^2 and t^2 −v^2 = s^2 −u^2 ⇒t^2 −y^2 = (t^2 −v^2 )−(y^2 −v^2 ) = (s^2 −u^2 )−(x^2 −u^2 ) = s^2 −x^2 i.e _((x,y)) R_((u,v)) and_((u,v)) R_((s,t)) ⇒_((x,y)) R_((s,t)) i.e transitive ∴ The relation is an equivalance relation.
R(x,y)⇒y2−y2=x2−x2⇒symmetricv2−y2=u2−x2⇔y2−v2=x2−u2⇒(x,y)R(u,v)⇒(u,v)R(x,y)i.ereflexivev2−y2=u2−x2andt2−v2=s2−u2⇒t2−y2=(t2−v2)−(y2−v2)=(s2−u2)−(x2−u2)=s2−x2i.e(x,y)R(u,v)and(u,v)R(s,t)⇒(x,y)R(s,t)i.etransitive∴Therelationisanequivalancerelation.

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