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Which of S (y=x+1) and T (x-y∈Z) is Equivalence | JEE

JEE Maths question with a full step-by-step solution.

Question

Let

R

be the real line. Consider the subsets

S = \{(x, y) : y = x + 1,\ 0 < x < 2\}

and

T = \{(x, y) : x - y \text{ is an integer}\}

of the plane

R \times R

. Which one of the following is true?

ABoth

S

and

T

are equivalence relations on

R

S

is an equivalence relation on

R

but

T

is not

T

is an equivalence relation on

R

but

S

is notcorrect

DNeither

S

nor

T

is an equivalence relation on

R

Solution

For

T \ (x - y \in \mathbb{Z})

x - x = 0 \in \mathbb{Z} \Rightarrow \text{reflexive}

x - y \in \mathbb{Z} \Rightarrow y - x \in \mathbb{Z} \Rightarrow \text{symmetric}

x - y,\ y - z \in \mathbb{Z} \Rightarrow x - z \in \mathbb{Z} \Rightarrow \text{transitive}

\therefore T

is an equivalence relation. For

S \ (y = x + 1)

x = x + 1

is impossible

\Rightarrow S

is not reflexive

\Rightarrow

not an equivalence relation. Correct answer: (3)

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