Q

# Solve it, - Sets, Relations and Functions - JEE Main-2

Let $\dpi{100} R$ be the real line. Consider the following subsets of the plane $\dpi{100} R \times R$ :

$\dpi{100} S=\left \{ (x,y):y=x+1\; and\; 0< x< 2 \right \}$

$\dpi{100} T=\left \{ (x,y):x-y\; is\;an\; integer \right \}$

Which one of the following is true?

• Option 1)

$T$ is an equivalence relation on $R$ but $S$ is not

• Option 2)

Neither $S$ nor $T$  is an equivalence relation on $R$

• Option 3)

Both $S$ and $T$ are equivalence relations on $R$

• Option 4)

$S$is an equivalence relation on $R$ but $T$ is not

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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)

$T$ is an equivalence relation on $R$ but $S$ is not

This is the correct option.

Option 2)

Neither $S$ nor $T$  is an equivalence relation on $R$

This is an incorrect option.

Option 3)

Both $S$ and $T$ are equivalence relations on $R$

This is an incorrect option.

Option 4)

$S$is an equivalence relation on $R$ but $T$ is not

This is an incorrect option.

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