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JEE MATHEMATICS MCQ: RELATIONS

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Analyzing +4 | -1
Let RR be the real line. Consider the following subsets of the plane R×RR \times R

S={(x,  y):y=x+1  and  o<x<2}S = \{ \left( {x,\;y} \right):y = x + 1\;and\;o < x < 2\}

T={(x,  y):x−yT = \{ \left( {x,\;y} \right):x - y is an integer}

Which of the following is true?
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Correct option: A
TT is an equivalent relation on RR but SS is not
Option A is correct because TT satisfies reflexive, symmetric, and transitive properties, making it an equivalence relation, while SS is not symmetric.

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In the set Z\mathrm{Z} of all integers, which of the following relation RR is not an equivalence relation? Let N\mathrm{N} denote the set of all natural numbers and R\mathrm{R} be the relation on N×N\mathrm{N} \times \mathrm{N} defined by (a,b)R(c,d)(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) if ad(b+c)=bc(a+d)a d(b+c)=b c(a+d), then R\mathrm{R} is Let WW denote the words in the English dictionary. Define the relation RR by: R={(x,y)∈W×W∣R=\{(x, y) \in W \times W \mid the words xx and yy have at least one letter in common }\} Let SS be the set of all real numbers. Then, the relation R={(a,  b):1+ab>0}R = \{ \left( {a,\;b} \right):1 + ab > 0\} on SS is Let XX be a family of sets and RR be a relation on XX defined by ' AA is disjoint from BB '. Then RR is: RR is relation on NN given by R={(x,  y):4x+3y=20}R = \left\{ {\left( {x,\;y} \right):4x + 3y = 20} \right\}. Which of the following belongs to RR?
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