Equivalence Relations

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JEE Main 2021 (February)

Let $R = {(P, Q) ∣ P and Q are at the same distance from the origin}$ be a relation, then the equivalence class of $(1, - 1)$ is the set:

S = {(x, y) ∣ x^{2} + y^{2} = 1}

(B)

S = {(x, y) ∣ x^{2} + y^{2} = 4}

(C)

S = {(x, y) ∣ x^{2} + y^{2} = 2}

(D)

S = {(x, y) ∣ x^{2} + y^{2} = 2}

JEE Main 2021 (March) Q1: 16 March (Shift 2)

LEVELJEE Main

Let $A = {2, 3, 4, 5, ..., 30}$ and ' $\sim$ ' be an equivalence relation on $A \times A$ , defined by $(a, b) \sim (c, d)$ , if and only if $a d = b c$ . Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4, 3)$ is equal to

JEE Main 2021 (March) Q3: 18 March (Shift 2)

LEVELJEE Main

Define a relation R over a class of $n \times n$ real matrices A and B as "ARB iff there exists a non-singular matrix P such that $P A P^{- 1} = B$ ". Then which of the following is true?

(A)

R is symmetric, transitive but not reflexive,

(B)

R is reflexive, symmetric but not transitive

(C)

R is an equivalence relation

(D)

R is reflexive, transitive but not symmetric

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

Let a set $A = A_{1} \cup A_{2} \cup \dots \cup A_{k}$ , where $A_{i} \cap A_{j} = ϕ$ for $i \neq = j, 1 \leq i, j \leq k$ . Define the relation $R$ from $A$ to $A$ by $R = {(x, y) : y \in A_{i} if and only if x \in A_{i}, 1 \leq i \leq k}$ . Then, $R$ is:

(A)

reflexive, symmetric but not transitive

(B)

reflexive, transitive but not symmetric

(C)

reflexive but not symmetric and transitive

(D)

an equivalence relation

JEE Main 2022 (27 July Shift 1)

LEVELJEE Main

Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R_{1} b ⟺ ab \geq 0$ and $a R_{2} b ⟺ a \geq b$ , then

(A)

R_{1}

is an equivalence relation but not

R_{2}

(B)

R_{2}

is an equivalence relation but not

R_{1}

(C)

both

R_{1}

and

R_{2}

are equivalence relations

(D)

neither

R_{1}

nor

R_{2}

is an equivalence relation

JEE Main 2023 (24 January Shift 2)

LEVELJEE Main

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c), (b, d)}$ on the set ${a, b, c, d}$ so that it is an equivalence relation, is ______.

JEE Main 2023 (29 January Shift 2)

LEVELBoard

Let $R$ be a relation defined on $N$ as $a R b$ is $2 a + 3 b$ is a multiple of 5, $a, b \in N$ . Then $R$ is

(A)

not reflexive

(B)

transitive but not symmetric

(C)

symmetric but not transitive

(D)

an equivalence relation

JEE Main 2023 (01 February Shift 2)

LEVELJEE Main

Let $P (S)$ denote the power set of $S = {1, 2, 3, \dots, 10}$ . Define the relations $R_{1}$ and $R_{2}$ on $P (S)$ as $A R_{1} B$ if $(A \cap B^{c}) \cup (B \cap A^{c}) = \emptyset$ and $A R_{2} B$ if $A \cup B^{c} = B \cup A^{c}, \forall A, B \in P (S)$ . Then:

(A)

both

R_{1}

and

R_{2}

are equivalence relations

(B)

only

R_{1}

is an equivalence relation

(C)

only

R_{2}

is an equivalence relation

(D)

both

R_{1}

and

R_{2}

are not equivalence relations

JEE Main 2025 (January)

LEVELJEE Main

Let $R = {(1, 2), (2, 3), (3, 3)}$ be a relation defined on the set ${1, 2, 3, 4}$ . Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

JEE Main 2025 (January)

LEVELBoard

The relation $R = {(x, y) : x, y \in Z and x + y is even}$ is:

(A)

reflexive and symmetric but not transitive

(B)

an equivalence relation

(C)

symmetric and transitive but not reflexive

(D)

reflexive and transitive but not symmetric

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

Let $A = {0, 1, 2, \dots, 9}$ . Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $∣ x - y ∣$ is a multiple of 3. Given below are two statements: Statement I: $n (R) = 36$ . Statement II: $R$ is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

(A)

(1) Statement I is incorrect but Statement II is correct

(B)

(2) Statement I is correct but Statement II is incorrect

(C)

(3) Both Statement I and Statement II are incorrect

(D)

(4) Both Statement I and Statement II are correct

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R={(P,Q)∣P and Q are at the same distance from the origin } be a relation, then the equivalence class of (1,−1) is the set:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Define a relation R over a class of n×n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP−1=B". Then which of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R1​ and R2​ be two relations defined on R by aR1​b⟺ab≥0 and aR2​b⟺a≥b, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The minimum number of elements that must be added to the relation R={(a,b),(b,c),(b,d)} on the set {a,b,c,d} so that it is an equivalence relation, is ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R be a relation defined on N as aRb is 2a+3b is a multiple of 5, a,b∈N. Then R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The relation R={(x,y):x,y∈Z and x+y is even } is:

Let $R = {(P, Q) ∣ P and Q are at the same distance from the origin}$ be a relation, then the equivalence class of $(1, - 1)$ is the set:

Define a relation R over a class of $n \times n$ real matrices A and B as "ARB iff there exists a non-singular matrix P such that $P A P^{- 1} = B$ ". Then which of the following is true?

Let $R_{1}$ and $R_{2}$ be two relations defined on $R$ by $a R_{1} b ⟺ ab \geq 0$ and $a R_{2} b ⟺ a \geq b$ , then

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c), (b, d)}$ on the set ${a, b, c, d}$ so that it is an equivalence relation, is ______.

Let $R$ be a relation defined on $N$ as $a R b$ is $2 a + 3 b$ is a multiple of 5, $a, b \in N$ . Then $R$ is

The relation $R = {(x, y) : x, y \in Z and x + y is even}$ is: