# Sets and Relations · Mathematics · JEE Main

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## Numerical

JEE Main 2024 (Online) 9th April Morning Shift
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 $$a_1+a_2=... JEE Main 2024 (Online) 4th April Morning Shift In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and... JEE Main 2024 (Online) 1st February Morning Shift Let A=\{1,2,3, \ldots, 20\}. Let R_1 and R_2 two relation on A such that R_1=\{(a, b): b is divisible by a\} R_2=\{(a, b): a is an inte... JEE Main 2024 (Online) 31st January Evening Shift Let$$A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$$. Let$$R$$be a relation on$$\mathrm{A}$$defined by$$(x, y) \in R$$if and only if$$2 x=3 y$... JEE Main 2024 (Online) 31st January Morning Shift Let $$A=\{1,2,3,4\}$$ and $$R=\{(1,2),(2,3),(1,4)\}$$ be a relation on $$\mathrm{A}$$. Let $$\mathrm{S}$$ be the equivalence relation on $$\mathrm{A}... JEE Main 2024 (Online) 30th January Evening Shift The number of symmetric relations defined on the set$$\{1,2,3,4\}$$which are not reflexive is _________. JEE Main 2023 (Online) 15th April Morning Shift The number of elements in the set \left\{n \in \mathbb{N}: 10 \leq n \leq 100\right. and 3^{n}-3 is a multiple of 7\} is ___________. JEE Main 2023 (Online) 15th April Morning Shift Let A=\{1,2,3,4\} and \mathrm{R} be a relation on the set A \times A defined by R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}. Then the number of ele... JEE Main 2023 (Online) 13th April Evening Shift Let$$\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$$and$$\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$$or$$\left.b^{2}=a+1\right\}$$... JEE Main 2023 (Online) 12th April Morning Shift The number of relations, on the set$$\{1,2,3\}$$containing$$(1,2)$$and$$(2,3)$$, which are reflexive and transitive but not symmetric, is _______... JEE Main 2023 (Online) 10th April Morning Shift The number of elements in the set$$\{ n \in Z:|{n^2} - 10n + 19| JEE Main 2023 (Online) 8th April Morning Shift 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 $$x-... JEE Main 2023 (Online) 6th April Morning Shift Let$$\mathrm{A}=\{1,2,3,4, \ldots ., 10\}$$and$$\mathrm{B}=\{0,1,2,3,4\}$$. The number of elements in the relation$$R=\left\{(a, b) \in A \times A... JEE Main 2023 (Online) 25th January Morning Shift Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________. JEE Main 2023 (Online) 24th January Evening Shift 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 re... JEE Main 2022 (Online) 29th July Morning Shift Let $$S=\{4,6,9\}$$ and $$T=\{9,10,11, \ldots, 1000\}$$. If $$A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\... JEE Main 2022 (Online) 26th July Evening Shift Let$$A=\{1,2,3,4,5,6,7\}$$and$$B=\{3,6,7,9\}$$. Then the number of elements in the set$$\{C \subseteq A: C \cap B \neq \phi\}$$is ___________.... JEE Main 2022 (Online) 25th July Evening Shift Let$$A=\{1,2,3,4,5,6,7\}$$. Define$$B=\{T \subseteq A$$: either$$1 \notin T$$or$$2 \in T\}$$and$$C=\{T \subseteq A: T$$the sum of all the ele... JEE Main 2022 (Online) 28th June Morning Shift Let R1 and R2 be relations on the set {1, 2, ......., 50} such that R1 = {(p, pn) : p is a prime and n$$\ge$$0 is an integer} and R2 = {(p, pn) : p ... JEE Main 2022 (Online) 26th June Morning Shift Let A = {n$$\in$$N : H.C.F. (n, 45) = 1} and Let B = {2k : k$$\in$${1, 2, ......., 100}}. Then the sum of all the elements of A$$\cap$$B is ____... JEE Main 2022 (Online) 26th June Morning Shift Let$$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$and$$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\ma... JEE Main 2022 (Online) 24th June Evening Shift The sum of all the elements of the set $$\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\}$$ is __________. JEE Main 2021 (Online) 27th August Morning Shift If A = {x $$\in$$ R : |x $$-$$ 2| > 1}, B = {x $$\in$$ R : $$\sqrt {{x^2} - 3}$$ > 1}, C = {x $$\in$$ R : |x $$-$$ 4| $$\ge$$ 2} and Z is the s... JEE Main 2021 (Online) 27th July Evening Shift Let A = {n $$\in$$ N | n2 $$\le$$ n + 10,000}, B = {3k + 1 | k$$\in$$ N} an dC = {2k | k$$\in$$N}, then the sum of all the elements of the set A $$\ca... JEE Main 2021 (Online) 24th February Morning Shift Let A = {n$$ \in $$N: n is a 3-digit number} B = {9k + 2: k$$ \in $$N} and C = {9k +$$l$$: k$$ \... JEE Main 2020 (Online) 6th September Morning Slot Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value o... JEE Main 2020 (Online) 7th January Evening Slot Let X = {n $$\in$$ N : 1 $$\le$$ n $$\le$$ 50}. If A = {n $$\in$$ X: n is a multiple of 2} and B = {n $$\in$$ X: n is a multiple of 7}, then... ## MCQ (Single Correct Answer) JEE Main 2024 (Online) 8th April Evening Shift 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 $$3 a d-7... JEE Main 2024 (Online) 6th April Evening Shift Let$$\mathrm{A}=\{1,2,3,4,5\}$$. Let$$\mathrm{R}$$be a relation on$$\mathrm{A}$$defined by$$x \mathrm{R} y$$if and only if$$4 x \leq 5 \mathrm... JEE Main 2024 (Online) 6th April Morning Shift Let $$A=\{n \in[100,700] \cap \mathrm{N}: n$$ is neither a multiple of 3 nor a multiple of 4$$\}$$. Then the number of elements in $$A$$ is JEE Main 2024 (Online) 6th April Morning Shift Let the relations $$R_1$$ and $$R_2$$ on the set $$X=\{1,2,3, \ldots, 20\}$$ be given by $$R_1=\{(x, y): 2 x-3 y=2\}$$ and $$R_2=\{(x, y):-5 x+4 y=0\}... JEE Main 2024 (Online) 4th April Evening Shift Let a relation$$\mathrm{R}$$on$$\mathrm{N} \times \mathbb{N}$$be defined as:$$\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$$if and only... JEE Main 2024 (Online) 1st February Evening Shift Consider the relations R_1 and R_2 defined as a R_1 b \Leftrightarrow a^2+b^2=1 for all a, b \in \mathbf{R} and (a, b) R_2(c, d) \Leftrightar... JEE Main 2024 (Online) 29th January Evening Shift If R is the smallest equivalence relation on the set$$\{1,2,3,4\}$$such that$$\{(1,2),(1,3)\} \subset \mathrm{R}$$, then the number of elements in ... JEE Main 2024 (Online) 29th January Morning Shift Let$$R$$be a relation on$$Z \times Z$$defined by$$(a, b) R(c, d)$$if and only if$$a d-b c$$is divisible by 5. Then$$R$$is JEE Main 2024 (Online) 27th January Evening Shift Let$$A$$and$$B$$be two finite sets with$$m$$and$$n$$elements respectively. The total number of subsets of the set$$A$$is 56 more than the to... JEE Main 2024 (Online) 27th January Morning Shift Let S=\{1,2,3, \ldots, 10\}. Suppose M is the set of all the subsets of S, then the relation \mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} ... JEE Main 2023 (Online) 11th April Evening Shift Let$$\mathrm{A}=\{1,3,4,6,9\}$$and$$\mathrm{B}=\{2,4,5,8,10\}$$. Let$$\mathrm{R}$$be a relation defined on$$\mathrm{A} \times \mathrm{B}$$such ... JEE Main 2023 (Online) 11th April Morning Shift An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals... JEE Main 2023 (Online) 10th April Evening Shift Let$$\mathrm{A}=\{2,3,4\}$$and$$\mathrm{B}=\{8,9,12\}$$. Then the number of elements in the relation$$\mathrm{R}=\left\{\left(\left(a_{1}, \mathr... JEE Main 2023 (Online) 8th April Evening Shift Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}$$ is : JEE Main 2023 (Online) 1st February Evening Shift Let $$P(S)$$ denote the power set of $$S=\{1,2,3, \ldots ., 10\}$$. Define the relations $$R_{1}$$ and $$R_{2}$$ on $$P(S)$$ as $$\mathrm{AR}_{1} \mat... JEE Main 2023 (Online) 1st February Morning Shift Let$$R$$be a relation on$$\mathbb{R}$$, given by$$R=\{(a, b): 3 a-3 b+\sqrt{7}$$is an irrational number$$\}$$. Then$$R$$is JEE Main 2023 (Online) 31st January Evening Shift Among the relations \mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right... JEE Main 2023 (Online) 31st January Morning Shift Let$$\mathrm{R}$$be a relation on$$\mathrm{N} \times \mathbb{N}$$defined by$$(a, b) ~\mathrm{R}~(c, d)$$if and only if$$a d(b-c)=b c(a-d)$$. Th... JEE Main 2023 (Online) 30th January Morning Shift The minimum number of elements that must be added to the relation$$ \mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}$$on the set$$\... JEE Main 2023 (Online) 29th January Evening Shift Let R be a relation defined on $$\mathbb{N}$$ as $$a\mathrm{R}b$$ if $$2a+3b$$ is a multiple of $$5,a,b\in \mathbb{N}$$. Then R is JEE Main 2023 (Online) 24th January Morning Shift The relation $$\mathrm{R = \{ (a,b):\gcd (a,b) = 1,2a \ne b,a,b \in \mathbb{Z}\}}$$ is : JEE Main 2022 (Online) 29th July Morning Shift Let R be a relation from the set $$\{1,2,3, \ldots, 60\}$$ to itself such that $$R=\{(a, b): b=p q$$, where $$p, q \geqslant 3$$ are prime numbers}. T... JEE Main 2022 (Online) 28th July Morning Shift For $$\alpha \in \mathbf{N}$$, consider a relation $$\mathrm{R}$$ on $$\mathbf{N}$$ given by $$\mathrm{R}=\{(x, y): 3 x+\alpha y$$ is a multiple of 7$...
JEE Main 2022 (Online) 27th July Morning Shift
Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by $$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrighta... JEE Main 2022 (Online) 29th June Morning Shift Let a set A = A1$$\cup$$A2$$\cup$$.....$$\cup$$Ak, where Ai$$\cap$$Aj =$$\phi$$for i$$\ne$$j, 1$$\le$$j, j$$\le$$k. Define the relatio... JEE Main 2022 (Online) 28th June Evening Shift Let R1 = {(a, b)$$\in$$N$$\times$$N : |a$$-$$b|$$\le$$13} and R2 = {(a, b)$$\in$$N$$\times$$N : |a$$-$$b|$$\ne$$13}. Then on N :... JEE Main 2021 (Online) 31st August Morning Shift Which of the following is not correct for relation R on the set of real numbers ? JEE Main 2021 (Online) 26th August Morning Shift Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are s... JEE Main 2021 (Online) 27th July Evening Shift Let N be the set of natural numbers and a relation R on N be defined by$$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\} $$. Then... JEE Main 2021 (Online) 18th March Evening Shift Define a relation R over a class of n$$\times$$n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP$$-$$1 = B". Th... JEE Main 2021 (Online) 17th March Morning Shift In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn ... JEE Main 2021 (Online) 16th March Evening Shift Let A = {2, 3, 4, 5, ....., 30} and '$$ \simeq $$' be an equivalence relation on A$$\times$$A, defined by (a, b)$$ \simeq $$(c, d), if and only if... JEE Main 2021 (Online) 16th March Morning Shift The number of elements in the set {x$$\in$$R : (|x|$$-$$3) |x + 4| = 6} is equal to : JEE Main 2021 (Online) 26th February Morning Shift Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1,$$-$$1) is the set : JEE Main 2020 (Online) 5th September Morning Slot A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coff... JEE Main 2020 (Online) 4th September Evening Slot Let$$\mathop \cup \limits_{i = 1}^{50} {X_i} = \mathop \cup \limits_{i = 1}^n {Y_i} = T$$where each Xi contains 10 elements and each Yi contains ... JEE Main 2020 (Online) 4th September Morning Slot A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a po... JEE Main 2020 (Online) 3rd September Evening Slot Let R1 and R2 be two relation defined as follows : R1 = {(a, b)$$ \in $$R2 : a2 + b2$$ \in $$Q} and R2 = {(a, b)$$ \in $$R2 : a2 + b2$$...
JEE Main 2020 (Online) 3rd September Morning Slot
Consider the two sets : A = {m $$\in$$ R : both the roots of x2 – (m + 1)x + m + 4 = 0 are real} and B = [–3, 5). Which of the following is not tr...
JEE Main 2020 (Online) 2nd September Morning Slot
If R = {(x, y) : x, y $$\in$$ Z, x2 + 3y2 $$\le$$ 8} is a relation on the set of integers Z, then the domain of R–1 is :...
JEE Main 2020 (Online) 9th January Evening Slot
If A = {x $$\in$$ R : |x| < 2} and B = {x $$\in$$ R : |x – 2| $$\ge$$ 3}; then :
JEE Main 2019 (Online) 12th April Evening Slot
Let A, B and C be sets such that $$\phi$$ $$\ne$$ A $$\cap$$ B $$\subseteq$$ C. Then which of the following statements is not true ?
JEE Main 2019 (Online) 9th April Evening Slot
Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Furt...
JEE Main 2019 (Online) 12th January Evening Slot
Let Z be the set of integers. If A = {x $$\in$$ Z : 2(x + 2) (x2 $$-$$ 5x + 6) = 1} and B = {x $$\in$$ Z : $$-$$ 3 < 2x $$-$$ 1 < 9}, t...
JEE Main 2019 (Online) 12th January Morning Slot
Let S = {1, 2, 3, … , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is :
JEE Main 2019 (Online) 10th January Morning Slot
In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics ...
JEE Main 2018 (Online) 16th April Morning Slot
Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y) $$\in$$ N $$\times$$ N : 2x + y = 10} and R2 = {(x, y...
JEE Main 2018 (Offline)
Two sets A and B are as under : A = {($$a$$, b) $$\in$$ R $$\times$$ R : |$$a$$ - 5| < 1 and |b - 5| < 1}; B = {($$a$$, b) $$\in$$ R $$\t... JEE Main 2018 (Online) 15th April Morning Slot Consider the following two binary relations on the set A = {a, b, c} : R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R2 = {(a, b), (b, a... JEE Main 2016 (Online) 10th April Morning Slot Let P = {$$\theta $$: sin$$\theta -$$cos$$\theta $$=$$\sqrt 2 \,\cos \theta $$} and Q = {$$\theta $$: sin$$\theta $$+ cos$$\theta $$=$$\...
JEE Main 2015 (Offline)
Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A $\times$ B , each having atleast three...
AIEEE 2012
Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$\subseteq$$ X, Z $$\subseteq$$ X and Y $$\... AIEEE 2011 Let R be the set of real numbers. Statement I : A=\{(x, y) \in R \times R: y-x is an integer \} is an equivalence relation on R. Statement II ... AIEEE 2010 Consider the following relations R=\{(x, y) \mid x, y are real numbers and x=w y for some rational number w\}; S=\left\{\left(\frac{m}{n}, \f... AIEEE 2009 If A, B and C are three sets such that A \cap B=A \cap C and A \cup B=A \cup C, then : AIEEE 2008 Let R be the real line. Consider the following subsets of the plane$$R \times R$$:$$S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right...
AIEEE 2006
Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least o...
AIEEE 2005
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 :
AIEEE 2004
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 :
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