If a relation R on the set \1, 2, 3\ be defined by R = \(1, 2)\, then R is (A) reflexive (B) transitive (C) sy

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Concept understanding — Relation Properties

Properties of a Relation

A relation $R$ on a set $A$ pairs elements of $A$ with one another. Some relations behave in regular, predictable ways, and we name these behaviours properties. Three matter most for CBSE Class 12 — reflexive, symmetric, transitive (together they build an equivalence relation); a fourth, antisymmetric, is worth knowing for order relations.

Reflexive — everything relates to itself

$R$ is reflexive if $a\,R\,a$ for every $a\in A$ . "Has the same age as" is reflexive; "is taller than" is not. If even one element misses its self-pair, reflexivity fails: on $\{1,2,3\}$ , $\{(1,1),(2,2)\}$ is not reflexive because $(3,3)$ is absent.

Symmetric — the relation runs both ways

$R$ is symmetric if $a\,R\,b \implies b\,R\,a$ . "Is married to" is symmetric; "is taller than" is not. Symmetry does not demand that every pair be related — only that any pair which appears also appears reversed. So $\{(1,2),(2,1),(3,3)\}$ is symmetric, but $\{(1,2),(2,1),(1,3)\}$ is not, since $(3,1)$ is missing.

Transitive — relations chain

$R$ is transitive if $a\,R\,b$ and $b\,R\,c$ together force $a\,R\,c$ . "Is an ancestor of" is transitive; "is a friend of" is not. A single broken chain breaks the property: $\{(1,2),(2,3)\}$ is not transitive because $(1,3)$ is missing.

Antisymmetric — two-way ties force equality

$R$ is antisymmetric if $a\,R\,b$ and $b\,R\,a$ together force $a=b$ . The order relation $\le$ is antisymmetric: $a\le b$ and $b\le a$ give $a=b$ . It does not ban self-pairs like $(1,1)$ ; it only rules out distinct elements related both ways.

Tip

Test the properties in order of ease — reflexivity first, then symmetry, transitivity. A single counterexample is enough to disprove any of them.

Property	Condition
Reflexive	$\forall a,\ a\,R\,a$

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On $\{1,2,3\}$ with $R=\{(1,2)\}$ : not reflexive, not symmetric, but vacuously transitive — option (B).

Check each property for $R=\{(1,2)\}$ on the set $\{1,2,3\}$ .

Reflexive? Needs $(1,1),(2,2),(3,3)\in R$ . None are present, so $R$ is not reflexive.

Symmetric? Since $(1,2)\in R$ , symmetry would require $(2,1)\in R$ . It is absent, so $R$ is not symmetric. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

GUJCET 2026Set x1 markMCQ
$R$
›Reveal solutionSolution
Test each pair against both conditions $a=b-2$ and $b<6$ .

$(6,8)$ : $b=8\not<6$ . ✗

$(8,7)$ : $b=7\not<6$ . ✗

$(8,3)$ : $a=b-2=1\ne8$ . ✗ …
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GSEB Higher Secondary Certificate (HSC) Examination 2025Set ANNUAL1 markMCQ
$R$
(a) $(2,4) \in R$
(b) $(6,8) \in R$
(c) $(3,8) \in R$
(d) $(8,7) \in R$
›Reveal solutionSolution
Test each pair against both conditions: $a=b-2$ and $b>6$ .

$(2,4)$ : $2=4-2$ holds, but $4>6$ fails.

$(6,8)$ : $6=8-2$ holds, and $8>6$ holds -- both conditions satisfied. …
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GSEB Higher Secondary Certificate (HSC) Examination 2024Set ANNUAL1 markMCQ
$R = \{(a,b),(b,a)\}$
(a) Reflexive, but not symmetric and transitive
(b) Symmetric, but not reflexive and transitive
(c) Transitive, but not reflexive and symmetric
(d) An equivalence relation
›Reveal solutionSolution

Check each property of $R=\{(a,b),(b,a)\}$ on $\{a,b,c\}$ directly against its definition.

Reflexive? Needs $(a,a),(b,b),(c,c)\in R$ -- none are present, so R is NOT reflexive.

Symmetric? $(a,b)\in R \Rightarrow (b,a)\in R$ -- both pairs are present, so R IS symmetric.

…

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GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
$\{1, 2, 3\}$
(a) R is an equivalence relation
(b) R is reflexive and symmetric but not transitive
(c) R is reflexive and transitive but not symmetric
(d) R is symmetric and transitive but not reflexive
›Reveal solutionSolution

Check the three properties on $R=\{(1,1),(2,2),(3,3)\}$ .

Reflexive: $(1,1),(2,2),(3,3)$ are all present ✓.

Symmetric: every pair is of the form $(a,a)$ , so $(a,a)\Rightarrow(a,a)$ ✓. …

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GSEB Higher Secondary Certificate (HSC) Examination 2022Set ANNUAL1 markMCQ
$*$
(a) $*$ is both associative and commutative
(b) $*$ is commutative but not associative
(c) $*$ is associative but not commutative
(d) $*$ is neither commutative nor associative
›Reveal solutionSolution

Test commutativity and associativity of $a*b=|a-b|$ .

Commutative: $a*b = |a-b| = |b-a| = b*a$ ✓.

Associative? $(a*b)*c = \big||a-b|-c\big|$ , while $a*(b*c) = \big|a-|b-c|\big|.$ …

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GSEB Higher Secondary Certificate (HSC) Examination 2020Set ANNUAL1 markMCQ
$R$
(a) $(2,4)\in R$
(b) $(3,8)\in R$
(c) $(6,8)\in R$
(d) $(8,7)\in R$
›Reveal solutionSolution
Check each ordered pair against both conditions of the relation: $a=b-2$ AND $b>6$ .

$R=\{(a,b): a=b-2,\ b>6\}$ on $N$ .

$(2,4)$ : $a=b-2 \Rightarrow 2=4-2$ ✓, but $b>6 \Rightarrow 4>6$ ✗. Rejected.

$(3,8)$ : $a=b-2 \Rightarrow 3=8-2=6$ ? ✗. Rejected. …
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GSEB Higher Secondary Certificate (HSC) Examination 2019Set ANNUAL1 markMCQ
$S = \{(1,1),(2,2),(3,3),(4,4),(5,5)\}$
(a) reflexive only
(b) symmetric only
(c) transistive only
(d) an equivalence relation
›Reveal solutionSolution

$S=\{(a,a): a\in\{1,...,5\}\}$ is the identity relation, and the identity relation is always an equivalence relation.

…

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GSEB Higher Secondary Certificate (HSC) Examination 2018Set ANNUAL1 markMCQ
$a * b = a^3 + b^3$
(a) $0$
(b) $729$
(c) $81$
(d) $27$
›Reveal solutionSolution

Evaluate the operation from the inside out.

Given $a*b=a^3+b^3$ on $\mathbb{Z}$ .

First $1*2=1^3+2^3=1+8=9$ .

…

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GSEB Higher Secondary Certificate (HSC) Examination 2018Set ANNUAL1 markMCQ
$R - \{0\}$
(a) an equivalence relation
(b) symmetric only
(c) reflexive only
(d) transitive only
›Reveal solutionSolution
"Same sign" is reflexive, symmetric and transitive on $R-\{0\}$ .

Reflexive: $x\cdot x=x^2>0$ for $x e 0$ , so $(x,x)\in S$ .

Symmetric: $xy>0\Rightarrow yx>0$ . …
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Q.If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$ , then $R$ is
(A) reflexive
(B) transitive
(C) symmetric
(D) none of these

Concept understanding — Relation Properties

Properties of a Relation

Reflexive — everything relates to itself

Symmetric — the relation runs both ways

Transitive — relations chain

Antisymmetric — two-way ties force equality

Method: Testing a small listed relation for R, S, T

Steps

Common Mistakes

Q.If a relation RRR on the set {1,2,3}\{1, 2, 3\}{1,2,3} be defined by R={(1,2)}R = \{(1, 2)\}R={(1,2)}, then RRR is (A) reflexive (B) transitive (C) symmetric (D) none of these

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Q.If a relation $R$ on the set $\{1, 2, 3\}$ be defined by $R = \{(1, 2)\}$ , then $R$ is
(A) reflexive
(B) transitive
(C) symmetric
(D) none of these

Unlock everything free for 14 days