Question

5. Show that if R is an equivalence relation on X then domR = rngR=X.​

Answer 1

SOLUTION

TO PROVE

If R is an equivalence relation on X then

dom R = rng R = X

EVALUATION

Here the given set is X

Let a ∈ dom X

Since R is equivalence relation

So there exists b ∈ X such that (a, b) ∈ R

Now (a, b) ∈ R

⇒ (b, a) ∈ R [ ∵ R is symmetric ]

Thus we have (a, b) ∈ R and (b, a) ∈ R

⇒ (a, a) ∈ R [ ∵ R is transitive ]

⇒ a ∈ rng X

Thus we have dom R ⊆ rng R

Similarly it can be shown that

rng R ⊆ dom R

Hence dom R = rng R = X

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