Sets and Relations

Consider the relation $R$ on the set $\{-2,-1,0,1,2\}$ defined by $(a, b) \in R$ if and only if $1+a b>0$. Then, among the statements :
I. The number of elements in R is 17
II. R is an equivalence relation

Let $\mathrm{R}=\left\{(x, y) \in \mathbf{N} \times \mathbf{N}: \log _{\mathrm{e}}(x+y) \leq 2\right\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is $\_\_\_\_$ .

Let $\mathrm{A}=\{1,4,7\}$ and $\mathrm{B}=\{2,3,8\}$. Then the number of elements, in the relation $R=\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right) \in((A \times B) \times(A \times B)): a_1+b_2\right.$ divides $\left.a_2+...

Let $A = \{2, 3, 4, 5, 6\}$. Let $R$ be a relation on the set $A \times A$ given by $(x, y)R(z, w)$ if and only if $x$ divides $z$ and $y \leq w$. Then the number of elements in $R$ is _________.

Let $R$ be a relation defined on the set $\{1,2,3,4\} \times\{1,2,3,4\}$ by
\(\mathrm{R}=\{((a, b),(c, d)): 2 a+3 b=3 c+4 d\} .\)
Then the number of elements in R is

Let $\mathrm{A}=\{-2,-1,0,1,2,3,4\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $2 x+y \leqslant 2$. Let $l$ be the number of elements in R . Let m and n be the minimum number of elements required to be added in R ...

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only if $|x-y|$ is a multiple of 3.
Given below are two statements :
Statement I : $n(\mathrm{R})=36$.
Statement II : R is an equivale...

Consider two sets $\mathrm{A}=\{x \in \mathrm{Z}:|(|x-3|-3)| \leq 1\}$ and
$\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _e(|x-2|)=0\right\}$. Then the number of
onto functions $f: \mathrm{A} \rightarrow \mathrm...

Let the relation R on the set $\mathrm{M}=\{1,2,3, \ldots, 16\}$ be given by $\mathrm{R}=\{(x, y): 4 y=5 x-3, x, y \in \mathrm{M}\}$.
Then the minimum number of elements required to be added in R , in order to make the relation symmetric, i...

Let S be the set of the first 11 natural numbers. Then the number of elements in $A=\{B \subseteq S: n(B) \geqslant 2$ and the product of all elements of $B$ is even $\}$ is $\_\_\_\_$ .

The number of elements in the relation $\mathrm{R}=\left\{(x, y): 4 x^2+y^2<52, x, y \in \mathbf{Z}\right\}$ is

The number of relations, defined on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$, which are both reflexive and symmetric, is equal to:

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \leq 3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric...

Let $A = \{x : |x^2 - 10| \leq 6\}$ and $B = \{x : |x - 2| > 1\}$. Then

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements (S1): The number of elements in R is 18, and (S2): The relation R is symmetric but neither reflexive nor...

For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2, \ldots, n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in S_6$, but $\{1,2,4\} \notin S_6$. Then $n\left(S_5\right)$ is equal to ________

The number of relations on the set $A=\{1,2,3\}$, containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is __________.

Let A = { ($\alpha, \beta$) $\in \mathbb{R} \times \mathbb{R}$ : |$\alpha$ - 1| $\leq 4$ and |$\beta$ - 5| $\leq 6$ } and B = { ($\alpha, \beta$) $\in \mathbb{R} \times \mathbb{R}$ : 16($\alpha$ - $2)^2 $+ 9($\beta$ - $6)^2$ $\leq 144$ }. T...

Consider the sets $A=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2=25\right\}, B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+9 y^2=144\right\}$, $C=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$ an...

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$ and R be a relation on A defined by $x \mathrm{R} y$ if and only if $2 x-y \in\{0,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added i...

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $0 \leq x^2+2 y \leq 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R t...

Let $A=\{-2,-1,0,1,2,3\}$. Let R be a relation on $A$ defined by $x \mathrm{R} y$ if and only if $y=\max \{x, 1\}$. Let $l$ be the number of elements in R . Let $m$ and $n$ be the minimum number of elements required to be added in R to make...

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is :

Let $A=\{1,2,3, \ldots ., 100\}$ and $R$ be a relation on $A$ such that $R=\{(a, b): a=2 b+1\}$. Let $\left(a_1\right.$, $\left.a_2\right),\left(a_2, a_3\right),\left(a_3, a_4\right), \ldots .,\left(a_k, a_{k+1}\right)$ be a sequence of $k$...

Define a relation R on the interval $ \left[0, \frac{\pi}{2}\right) $ by $ x $ R $ y $ if and only if $ \sec^2x - \tan^2y = 1 $. Then R is :

Let $\mathrm{S}=\mathbf{N} \cup\{0\}$. Define a relation R from S to $\mathbf{R}$ by :
\(\mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\} .\)
Then, th...

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

Let $S=\left\{p_1, p_2 \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y), x \in S$, $y \i...

Let $\mathrm{A}=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 /\pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$ and $\mathrm{B}=\{x \geqslant 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $\mathrm{n}(\mathrm{A} \cup ...

Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ 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:

Let $\mathrm{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x+y| \geqslant 3\}$ and $\mathrm{B}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x|+|y| \leq 3\}$.
If $\mathrm{C}=\{(x, y) \in \mathrm{A} \cap \mathrm{B}: x=0$ or $y=0\}$, then $\sum_...

Let $\mathrm{X}=\mathbf{R} \times \mathbf{R}$. Define a relation R on X as :
\(\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2\)
Statement I: $\quad \mathrm{R}$ is an equivalence relation.
Statement II : For some $(\mat...

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m< n\right.$ and $\left.\operatorname{gcd}(m, n)=1\right\}$. Then $n(B)$ is equal to :

Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is _________.

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 __________.

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

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

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 \(R_2=\{(x, y):-5 x+4 y=0\}\). If \(M\) and \(N\) be the minimum number of elements required to be added in \(R_1\) a...

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 \(4 x \leq 5 \mathrm{y}\). Let \(\mathrm{m}\) be the number of elements in \(\mathrm{R}\) and \(\mathrm{n}\) b...

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 at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both...

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 if \(x_1 \leq x_2\) or \(y_1 \leq y_2\).
Consider the two statements:
(I) \(\mathrm{R}\) ...

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 \(\mathrm{A}\) such that \(R \subset S\) and the number of elements in \(\mathrm{S}\) is \(\mathrm{n}\)...

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 \(2 x=3 y\). Let \(R_1\) be a symmetric relation on \(A\) such that \(R \subset R_1\) and the number...

The number of symmetric relations defined on the set \(\{1,2,3,4\}\) which are not reflexive is _________.

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

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 \(\mathrm{R}\) is __________.

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} \cap \mathrm{B} \neq \phi ; \mathrm{A}, \mathrm{B} \in \mathrm{M}\}$ is :

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 total number of subsets of \(B\). Then the distance of the point \(P(m, n)\) from the point ...

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 integral multiple of $b\}$.
Then, number of elements in $R_1-R_2$ is equal to _____________.