Sets and Relations

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 $(a, b) R_2(c, d) \Leftrightarrow$ $a+d=b+c$ for all $(a, b),(c, d) \in \mathbf{N} \times \mathbf{N}$. Then :

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 ...

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 :

Let \(\mathrm{A}=\{1,2,3,4, \ldots ., 10\}\) and \(\mathrm{B}=\{0,1,2,3,4\}\). The number of elements in the relation \(R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}\) is ___________.

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

Among the relations

$\mathrm{S}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R}-\{0\}, 2+\frac{\mathrm{a}}{\mathrm{b}}>0\right\}$
and $\mathrm{T}=\left\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathbb{R...

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 \(\{a, b, c\}\) so that it becomes symmetric and transitive is :

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

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 _____________.

The relation \(\mathrm{R = \{ (a,b):\gcd (a,b) = 1,2a \ne b,a,b \in \mathbb{Z}\}}\) is :

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 on \(\mathbb{R}\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number \(\}\). Then \(R\) is

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 \(\mathrm{AR}_{1} \mathrm{~B}\) if $$\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \...

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 elements in $\mathrm{R}$ is ____________.

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 ___________.

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 \(\left.b^{2}=a+1\right\}\) be a relation on \(\mathrm{A}\). Then the minimum number of elements, that must be added ...

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 __________.

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 in all the three events, then, how many received medals in exactly two of three events?

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 that $$\mathrm{R}=\left\{\left(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\right):...

The number of elements in the set \(\{ n \in Z:|{n^2} - 10n + 19| < 6\}\) is _________.

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}, \mathrm{~b}_{1}\right),\left(a_{2}, \mathrm{~b}_{2}\right)\right) \in(A \times B, A \times B): a...

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 relation R from A to A by R = {(x, y) : y \(\in\) Ai if and only if x \(\in\) Ai, 1 \(\le\) i $$\...

Let \(S=\{4,6,9\}\) and \(T=\{9,10,11, \ldots, 1000\}\). If \(A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in \mathbf{N}, a_{1}, a_{2}, a_{3}, \ldots, a_{k}\right.\) \(\epsilon S\}\), then the sum of all the elements in the set \(T-A\) is equal to...

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}. Then, the number of elements in R is :

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 is a prime and n = 0 or 1}.
Then, the number of elements in R1 \(-\) R2 is _______________...

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 :

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\(\}\). The relation \(R\) is an equivalence relation if and only if :

Let \(R_{1}\) and \(R_{2}\) be two relations defined on \(\mathbb{R}\) by
\(a \,R_{1} \,b \Leftrightarrow a b \geq 0\) and \(a \,R_{2} \,b \Leftrightarrow a \geq b\)
Then,

Let \(A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } }\) and \(B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } }\). Then A + B is equal to _____________.

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 ____________.

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 ___________.

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 elements of \(T\) is a prime number \(\}\). Then the number of elements in the set $$B \cup C...

The sum of all the elements of the set \(\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\}\) is __________.

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

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 \(\cap\)(B \(-\) C) is equal to _____________.

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 :

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 set of all integers, then the number of subsets of the set (A \(\cap\) B \(\cap\) C)c \(\cap\) Z ...

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 :

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 suffering from both ailments, then K can not belong to the set :

Let  A = {n \(\in\) N: n is a 3-digit number}
       B = {9k + 2: k \(\in\) N}
and C = {9k + \(l\): k \(\in\) N} for some \(l ( 0 < l < 9)\)
If the sum of all the elements of the set A \(\cap\) (B \(\cup\) C) is 274 \(\times\) 4...

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". Then which of the following is true?

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 diagrams can justify the above statement?

The number of elements in the set {x \(\in\) R : (|x| \(-\) 3) |x + 4| = 6} is equal to :

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 ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with or...

If A = {x \(\in\) R : |x| < 2} and B = {x \(\in\) R : |x – 2| \(\ge\) 3};
then :

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 the number of elements in the smallest subset of X
containing both A and B is ________.

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 of m.n is ______.

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 coffee and tea, then x cannot be :

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 possible value of x can be:

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 5 elements. If each element of the set T is an element of exactly 20 of sets Xi’s and exac...