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Prove that intersection of equivalence relations on a set is a

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

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Text solutionVerified

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

R \cap S

is also equivalence

x ϵ A \Rightarrow (x, x ϵ R)

and

(x, x ϵ S)

(x, x) ϵ R \cap S

∴ R \cap S

is Reflexive

Now

(x, y) ϵ R \cap S

(x, y) ϵ R

and

(x, y) ϵ S

(x, y) ϵ R

are equivalence they are symmetric also.

(y, x) ϵ R

and

(y, x) ϵ S .

(y, x) ϵ R \cap S

∴ R \cap S

is symmetric.

finally

(x, y) ϵ R \cap S

and

(y, z) ϵ R \cap S

(x, y) ϵ R

and

(y, z) ϵ R \Rightarrow (x, z) ϵ R

(x, y) ϵ S

and

(y, z) ϵ S \Rightarrow (x, z) ϵ S

∴ R

is transitive and 's' is transitive

∴ R \cap S

is transitive

∴ R \cap S

is equivalence relation.

i.e, Intersection of equivalence relation set

is equivalence.

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Question Text	Prove that intersection of equivalence relations on a set is also an equivalence relation.
Answer Type	Text solution:1
Upvotes	151

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

Text solutionVerified

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