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Question

Prove that intersection of equivalence relations on a set is also an equivalence relation.

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Solution

We have to prove intersection of
equivalence relations on a set is equivalence
Let us suppose R and S are 2 equivalence
relations on a set A.
So set A implies RS is also equivalence
xϵA(x,xϵR) and (x,xϵS)
(x,x)ϵRS
RS is Reflexive
Now (x,y)ϵRS
(x,y)ϵR and (x,y)ϵS
(x,y)ϵR are equivalence they are symmetric also.
(y,x)ϵR and (y,x)ϵS.
(y,x)ϵRS
RS is symmetric.
finally (x,y)ϵRS and (y,z)ϵRS
(x,y)ϵR and (y,z)ϵR(x,z)ϵR
(x,y)ϵS and (y,z)ϵS(x,z)ϵS
R is transitive and 's' is transitive
RS is transitive
RS is equivalence relation.
i.e, Intersection of equivalence relation set
is equivalence.


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