Types of Relations

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JEE Main 2011

LEVELJEE Main

Let R be the set of real numbers. Statement-1: $A = {(x, y) \in R \times R : y - x is an integer}$ is an equivalence relation on R. Statement-2: $B = {(x, y) \in R \times R : x = α y for some rational number α}$ is an equivalence relation on R.

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2010

LEVELJEE Main

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S = {( $\frac{m}{n}, \frac{p}{q}$ ) | m, n, p and q are integers such that $n, q \neq = 0$ and qm = pn}. Then

(A)

Neither R nor S is an equivalence relation

(B)

S is an equivalence relation but R is not an equivalence relation

(C)

R and S both are equivalence relations

(D)

R is an equivalence relation but S is not an equivalence relation

JEE Main 2008

LEVELJEE Main

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$ : $S = {(x, y) : y = x + 1 and 0 < x < 2}$ , $T = {(x, y) : x - y is an integer}$ . Which one of the following is true?

(A)

Neither S nor T is an equivalence relation on R

(B)

Both S and T are equivalence relation on R

(C)

S is an equivalence relation on R but T is not

(D)

T is an equivalence relation on R but S is not

JEE Main 2006

LEVELJEE Main

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) \in W \times W ∣ the words x and y have at least one letter in common.}$ Then R is

(A)

not reflexive, symmetric and transitive

(B)

reflexive, symmetric and not transitive

(C)

reflexive, symmetric and transitive

(D)

reflexive, not symmetric and transitive

JEE Main 2005

LEVELBoard

Let $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)}$ be a relation on the set $A = {3, 6, 9, 12}$ . The relation is

(A)

reflexive and transitive only

(B)

reflexive only

(C)

an equivalence relation

(D)

reflexive and symmetric only

JEE Main 2004

LEVELBoard

Let $R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)}$ be a relation on the set $A = {1, 2, 3, 4}$ . The relation R is

JEE Main 2018 (15 April Shift 1)

LEVELBoard

Consider the following two binary relations on the set A={a,b,c}: $R_{1} = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)}$ and $R_{2} = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}$ . Then

(A)

R_{1}

is not symmetric but it is transitive.

(B)

both

R_{1}

and

R_{2}

are transitive

(C)

R_{2}

is symmetric but it is not transitive

(D)

both

R_{1}

and

R_{2}

are not symmetric

JEE Main 2020 (3 September Shift 2)

LEVELJEE Main

Let $R_{1}$ and $R_{2}$ be two relation defined as follows: $R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}$ and $R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \in / Q}$ , where $Q$ is the set of all rational numbers. Then:

(A)

Neither

R_{1}

nor

R_{2}

is transitive.

(B)

R_{2}

is transitive but

R_{1}

is not transitive.

(C)

R_{1}

and

R_{2}

are both transitive.

(D)

R_{1}

is transitive but

R_{2}

is not transitive.

JEE Main 2021 (31 Aug Shift 1)

LEVELJEE Main

Which of the following is not correct for relation $R$ on the set of real numbers?

(A)

(x, y) \in R \Leftrightarrow 0 < ∣ x ∣ - ∣ y ∣ \leq 1

is neither transitive nor symmetric.

(B)

(x, y) \in R \Leftrightarrow 0 < ∣ x - y ∣ \leq 1

is symmetric and transitive.

(C)

(x, y) \in R \Leftrightarrow ∣ x ∣ - ∣ y ∣ \leq 1

is reflexive but not symmetric.

(D)

(x, y) \in R \Leftrightarrow ∣ x - y ∣ \leq 1

is reflexive and symmetric.

JEE Main 2021 (27 July Shift 2)

LEVELJEE Main

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = {(x, y) \in N \times N : x^{3} - 3 x^{2} y - x y^{2} + 3 y^{3} = 0}$ . Then the relation $R$ is :

(A)

symmetric but neither reflexive nor transitive

(B)

reflexive but neither symmetric nor transitive

(C)

reflexive and symmetric, but not transitive

(D)

an equivalence relation

JEE Main 2022 (28 June Shift 2)

LEVELJEE Main

Let $R_{1} = {(a, b) \in N \times N : ∣ a - b ∣ \leq 13}$ and $R_{2} = {(a, b) \in N \times N : ∣ a - b ∣ \neq = 13}$ . Then on $N$ :

(A)

Both

R_{1}

and

R_{2}

are equivalence relations

(B)

Neither

R_{1}

nor

R_{2}

is an equivalence relation

(C)

R_{1}

is an equivalence relation but

R_{2}

is not

(D)

R_{2}

is an equivalence relation but

R_{1}

is not

JEE Main 2022 (29 June Shift 2)

LEVELJEE Main

The probability that a relation $R$ from ${x, y}$ to ${x, y}$ is both symmetric and transitive, is equal to:

(A)

\frac{5}{16}

(B)

\frac{9}{16}

(C)

\frac{11}{16}

(D)

\frac{13}{16}

JEE Main 2023 (08 Apr Shift 1)

LEVELJEE Main

Let $A = {0, 3, 4, 6, 7, 8, 9, 10}$ and $R$ be the relation defined on $A$ such that $R = {(x, y) \in A \times A : x - y is odd positive integer or x - y = 2}$ . The minimum number of elements that must be added to the relation $R$ , so that it is a symmetric relation, is equal to

JEE Main 2023 (08 Apr Shift 2)

LEVELBoard

Let $A = {1, 2, 3, 4, 5, 6, 7}$ . Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is

(A)

an equivalence relation

(B)

symmetric but neither reflexive nor transitive

(C)

transitive but neither symmetric nor reflexive

(D)

reflexive but neither symmetric nor transitive

JEE Main 2023 (13 Apr Shift 2)

LEVELJEE Main

Let $A = {- 4, - 3, - 2, 0, 1, 3, 4}$ and $R = {(a, b) \in A \times A : b = ∣ a ∣ or b^{2} = a + 1}$ be a relation on $A$ . Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is

JEE Main 2023 (24 January Shift 1)

LEVELJEE Main

The relation $R = {(a, b) : g cd (a, b) = 1, 2 a \neq = b, a, b \in Z}$ is:

(A)

transitive but not reflexive

(B)

symmetric but not transitive

(C)

reflexive but not symmetric

(D)

neither symmetric nor transitive

JEE Main 2023 (30 January Shift 1)

LEVELJEE Main

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is:

JEE Main 2023 (31 January Shift 1)

LEVELJEE Main

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d (b - c) = b c (a - d)$ . Then $R$ is

(A)

symmetric but neither reflexive nor transitive

(B)

transitive but neither reflexive nor symmetric

(C)

reflexive and symmetric but not transitive

(D)

symmetric and transitive but not reflexive

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

Among the relations $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ ,

(A)

S is transitive but T is not

(B)

T is symmetric but S is not

(C)

Neither S nor T is transitive

(D)

Both S and T are symmetric

JEE Main 2023 (01 February Shift 1)

LEVELJEE Main

Let $R$ be a relation on $R$ , given by $R = {(a, b) : 3 a - 3 b + 7 is an irrational number}$ . Then $R$ is

(A)

Reflexive but neither symmetric nor transitive

(B)

Reflexive and transitive but not symmetric

(C)

Reflexive and symmetric but not transitive

(D)

An equivalence relation

JEE Main 2024 (04 Apr Shift 2)

LEVELJEE Main

Let a relation $R$ on $N \times N$ be defined as: $(x_{1}, y_{1}) R (x_{2}, y_{2})$ if and only if $x_{1} \leq x_{2}$ or $y_{1} \leq y_{2}$ . Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

(A)

(1) Both (I) and (II) are correct.

(B)

(2) Only (II) is correct.

(C)

(3) Neither (I) nor (II) is correct.

(D)

(4) Only (I) is correct.

JEE Main 2024 (06 Apr Shift 1)

LEVELJEE Main

Let the relations $R_{1}$ and $R_{2}$ on the set $X = {1, 2, 3, \dots, 20}$ be given by $R_{1} = {(x, y) : 2 x - 3 y = 2}$ and $R_{2} = {(x, y) : - 5 x + 4 y = 0}$ . If $M$ and $N$ be the minimum number of elements required to be added in $R_{1}$ and $R_{2}$ , respectively, in order to make the relations symmetric, then $M + N$ equals

JEE Main 2024 (06 Apr Shift 2)

LEVELJEE Main

Let $A = {1, 2, 3, 4, 5}$ . Let $R$ be a relation on $A$ defined by $x R y$ if and only if $4 x \leq 5 y$ . Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A \times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m + n$ is equal to :

JEE Main 2024 (08 Apr Shift 2)

LEVELJEE Main

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is

(A)

(1) an equivalence relation.

(B)

(2) reflexive and symmetric but not transitive.

(C)

(3) transitive but not symmetric.

(D)

(4) reflexive but not symmetric.

JEE Main 2024 (09 Apr Shift 1)

LEVELJEE Main

Let $A = {2, 3, 6, 7}$ and $B = {4, 5, 6, 8}$ . Let $R$ be a relation defined on $A \times B$ by $(a_{1}, b_{1}) R (a_{2}, b_{2})$ if and only if $a_{1} + a_{2} = b_{1} + b_{2}$ . Then the number of elements in $R$ is _________

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R be the real line. Consider the following subsets of the plane R×R: S={(x,y):y=x+1 and 0<x<2}, T={(x,y):x−y is an integer}. Which one of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let W denote the words in the English dictionary. Define the relation R by R={(x,y)∈W×W∣the words x and y have at least one letter in common.} Then R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R={(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)} be a relation on the set A={3,6,9,12}. The relation is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R={(1,3),(4,2),(2,4),(2,3),(3,1)} be a relation on the set A={1,2,3,4}. The relation R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the following two binary relations on the set A={a,b,c}: R1​={(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)} and R2​={(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)}. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R1​ and R2​ be two relation defined as follows: R1​={(a,b)∈R2:a2+b2∈Q} and R2​={(a,b)∈R2:a2+b2∈/Q}, where Q is the set of all rational numbers. Then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following is *not* correct for relation R on the set of real numbers?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let N be the set of natural numbers and a relation R on N be defined by R={(x,y)∈N×N:x3−3x2y−xy2+3y3=0}. Then the relation R is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R1​={(a,b)∈N×N:∣a−b∣≤13} and R2​={(a,b)∈N×N:∣a−b∣=13}. Then on N:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability that a relation R from {x,y} to {x,y} is both symmetric and transitive, is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={1,2,3,4,5,6,7}. Then the relation R={(x,y)∈A×A:x+y=7} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The relation R={(a,b):gcd(a,b)=1,2a=b,a,b∈Z} is:

.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)} on the set {a,b,c} so that it becomes symmetric and transitive is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R be a relation on N×N defined by (a,b)R(c,d) if and only if ad(b−c)=bc(a−d). Then R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Among the relations S={(a,b):a,b∈R−{0},2+ba​>0} and T={(a,b):a,b∈R,a2−b2∈Z},

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R be a relation on R, given by R={(a,b):3a−3b+7​ is an irrational number}. Then R is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={2,3,6,8,9,11} and B={1,4,5,10,15}. Let R be a relation on A×B defined by (a,b)R(c,d) if and only if 3ad−7bc is an even integer. Then the relation R is

Let $R$ be the real line. Consider the following subsets of the plane $R \times R$ : $S = {(x, y) : y = x + 1 and 0 < x < 2}$ , $T = {(x, y) : x - y is an integer}$ . Which one of the following is true?

Let W denote the words in the English dictionary. Define the relation R by $R = {(x, y) \in W \times W ∣ the words x and y have at least one letter in common.}$ Then R is

Let $R = {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)}$ be a relation on the set $A = {3, 6, 9, 12}$ . The relation is

Let $R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)}$ be a relation on the set $A = {1, 2, 3, 4}$ . The relation R is

Consider the following two binary relations on the set A={a,b,c}: $R_{1} = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)}$ and $R_{2} = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}$ . Then

Let $R_{1}$ and $R_{2}$ be two relation defined as follows: $R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}$ and $R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \in / Q}$ , where $Q$ is the set of all rational numbers. Then:

Which of the following is not correct for relation $R$ on the set of real numbers?

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = {(x, y) \in N \times N : x^{3} - 3 x^{2} y - x y^{2} + 3 y^{3} = 0}$ . Then the relation $R$ is :

Let $R_{1} = {(a, b) \in N \times N : ∣ a - b ∣ \leq 13}$ and $R_{2} = {(a, b) \in N \times N : ∣ a - b ∣ \neq = 13}$ . Then on $N$ :

The probability that a relation $R$ from ${x, y}$ to ${x, y}$ is both symmetric and transitive, is equal to:

Let $A = {1, 2, 3, 4, 5, 6, 7}$ . Then the relation $R = {(x, y) \in A \times A : x + y = 7}$ is

The relation $R = {(a, b) : g cd (a, b) = 1, 2 a \neq = b, a, b \in Z}$ is:

The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is:

Let $R$ be a relation on $N \times N$ defined by $(a, b) R (c, d)$ if and only if $a d (b - c) = b c (a - d)$ . Then $R$ is

Among the relations $S = {(a, b) : a, b \in R - {0}, 2 + \frac{a}{b} > 0}$ and $T = {(a, b) : a, b \in R, a^{2} - b^{2} \in Z}$ ,

Let $R$ be a relation on $R$ , given by $R = {(a, b) : 3 a - 3 b + 7 is an irrational number}$ . Then $R$ is

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$ . Let $R$ be a relation on $A \times B$ defined by $(a, b) R (c, d)$ if and only if $3 a d - 7 b c$ is an even integer. Then the relation $R$ is