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?a)The relation is an equivalent relation.b)The relation is not reflexive but it is symmetric and transitive.c)The relation is not symmetric but is reflexive and transitive.d)The relation is not transitive but it is reflexive and symmetric.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

Equivalence Relation:
An equivalence relation is a relation that satisfies three properties:

1. Reflexive: For all x ∈ S, xRx
2. Symmetric: For all x, y ∈ S, if xRy then yRx
3. Transitive: For all x, y, z ∈ S, if xRy and yRz then xRz

Let's check whether the given relation is an equivalence relation or not:

a) Reflexive: For all x ∈ S, xRx (If x was born in Delhi on a particular day, then x was born in Delhi on that day. So, x is related to x. Hence, the relation is reflexive.)

b) Symmetric: For all x, y ∈ S, if xRy then yRx (If x and y were born in Delhi on the same day, then y and x were also born in Delhi on the same day. Hence, the relation is symmetric.)

c) Transitive: For all x, y, z ∈ S, if xRy and yRz then xRz (If x and y were born in Delhi on the same day and y and z were also born in Delhi on the same day, then x and z were also born in Delhi on the same day. Hence, the relation is transitive.)

As the given relation satisfies all three properties of an equivalence relation, the correct option is A, i.e., the relation is an equivalence relation.

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