Let S be the set of all persons living in Delhi. We say that x, y in S are related if they were born in Delhi on the same day. Which one of the following is correct?

Concept:

Definition of Equivalence relation

• A relation is an equivalence relation if & only if it is:
• Reflexive property: aRa
• Symmetric property: if a R b, then b R a
• Transitive property: if a R b and b R c, then a R c

Calculation:

Given that,

S = persons living in Delhi x, y ∈ S.

x R y if x & y born on same day.

⇒ First we check reflexive property so,

x R x or y R y both will be reflective.

So reflexive property true for Relation.

⇒ Now symmetric property,

Given that x R y born on same day so it’s reverse y R x also true y and x also born on same day.

Symmetric property also true for relation.

⇒ For transitive property,

Given that x R y born on same day

Let’s take z who also born on same day of y so y R z ∈ S

So we can say that x R z ∈ S because given that x R y and y R z ∈ S.

Transitive property also true for relation.