#### Let be the real line. Consider the following subsets of the plane :Which one of the following is true? Option 1) is an equivalence relation on but is not Option 2) Neither nor   is an equivalence relation on Option 3) Both and are equivalence relations on Option 4) is an equivalence relation on but is not

As we learnt in

REFLEXIVE RELATION -

A relation R in A is said to be reflexive,  if a R a ,∀ a ∈ A

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SYMMETRIC RELATION -

A relation R in A is said to be symmetric, if a R b ⇒ b R a,∀ a,b ∈ A

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TRANSITIVE RELATION -

A relation R in A is said to be transitive, if a R b and b R c ⇒ a R c ∀ a,b,c ∈ A

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S = {y = x + 1        0 < x < 2}

T = {x - y is an integer}

For T :     (a, a) it is integer reflexive

(a, b) (b, a) it is integer symmetric.

(a, b) (b, c) (a, c) it is integer transitive.

So it is equivalence

For S : x = 1,  y = 2 not equivalence.

Correct option is 1.

Option 1)

is an equivalence relation on but is not

This is the correct option.

Option 2)

Neither nor   is an equivalence relation on

This is an incorrect option.

Option 3)

Both and are equivalence relations on

This is an incorrect option.

Option 4)

is an equivalence relation on but is not

This is an incorrect option.