Relations

Let $S=\{1,2,3,\dots,10\}$. Consider the set
$$X=\{R:R\text{ is an equivalence relation on the set }S\text{ such that }R\text{ has exactly 42 elements}\}.$$
Then the number of elements in $X$ is ________.

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 $y\le w$. Then the number of elements in R is ________.

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

Let $R=\left\{(x,y)\in\mathbf N\times\mathbf N:\log_e(x+y)\le2\right\}$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is ________.

Consider the relation R on the set $\{-2,-1,0,1,2\}$ defined by $(a,b)\in R$ if and only if $1+ab>0$. Then, among the statements :

I. The number of elements in R is 17

II. R is an equivalence relation

Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal{S}$.

Then the number of elements in $\mathcal{S}$ is _____.

If $n(X) = {}^{m}C_6$, then the value of $m$ is _____.

If the value of $n(Y) + n(Z)$ is $k^2$, then $|k|$ is _____.

Let $R_1$ and $R_2$ be relations on the set $\{1,2,\dots,50\}$ such that

$R_1=\left\{\left(p,p^n\right):p\text{ is a prime and }n\ge0\text{ is an integer}\right\}$ and

$R_2=\left\{\left(p,p^n\right):p\text{ is a prime and }n=0\text{ or }1\right\}$.

Then, the number of elements in $R_1-R_2$ is ______.

Let $R_1=\{(a,b)\in\mathbf N\times\mathbf N:|a-b|\le13\}$ and $R_2=\{(a,b)\in\mathbf N\times\mathbf N:|a-b|\ne13\}$. Then on $\mathbf N$ :

Let a set $A=A_1\cup A_2\cup\dots\cup A_k$, where $A_i\cap A_j=\phi$ for $i\ne j$, $1\le i,j\le k$. Define the relation R from A to A by $R=\{(x,y):y\in A_i\text{ if and only if }x\in A_i,\ 1\le i\le k\}$. Then, R is :

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

Let $R_1$ and $R_2$ be two relations defined on $\mathbf R$ by

$a\,R_1\,b\Leftrightarrow ab\ge0$ and $a\,R_2\,b\Leftrightarrow a\ge b$

Then,

For $\alpha\in\mathbf N$, consider a relation R on $\mathbf N$ given by $R=\{(x,y):3x+\alpha y\text{ is a multiple of }7\}$. The relation R is an equivalence relation if and only if :

Let R be a relation from the set {1, 2, 3, . . . , 60} to itself such that R = {(a, b) : b = pq, where p, q ⩾ 3 are prime numbers}. Then, the number of elements in R is :

Let P be the relation defined on the set of all real numbers such that
$$P=\left\{(a,b):\sec^2a-\tan^2b=1\right\}.$$
Then P is :

A relation on the set $A=\{x:|x|<3,\ x\in Z\}$, where $Z$ is the set of integers is defined by $R=\{(x,y):y=|x|,\ x\ne-1\}$. Then the number of elements in the power set of R is :