Basics of Relations

If $A$ is non-empty, then any subset of $A\times A$ is called 

A relation on set $A$ is a subset of 

If $A$ is finite set containing $m$ elements, then the number of relations from $A$ to $A$ is equal to 

If $A$ and $B$ are finite sets containing $m$ and $n$ elements respectively, then the number of relations from $A$ to $B$ is equal to 

If $A=\left\{1,-1\right\}$, $B=\left\{2,5\right\}$, then the number of relationships from $A$ to $B$ is

Let $A=\left\{2,3,4\right\}$ and $B=\left\{8,9,12\right\}$. Then the number of elements in the relation  $R=\left\{\left(\left(a_1,b_1\right),\left(a_2,b_2\right)\right)\in \left(A\times B,A\times B\right):a_1 \text{divides} b_2 \text{and} a_2 \text{divides} b_1\right\}$ is

Assertion $\left(\mathrm{A}\right)$: Domain and range of a relation $R=\left\{\left(x,y\right):x-3y=0\right\}$ defined on the set $\mathrm{A}=\left\{1,2,3\right\}$ are respectively $\left\{1,2,3\right\} \mathrm{and} \left\{2,4,6\right\}$.

Reason $\left(\mathrm{R}\right)$: Domain and Range of a relation $\mathrm{R}$ are respectively the sets $\left\{a:a\in A \mathrm{and} \left(a,b\right)\in R\right\} \mathrm{and} \left\{b:b\in A \mathrm{and} \left(a,b\right)\in R\right\}$

Let $n\left(A\right)=m$ and $n\left(B\right)=n$, then the total number of non-empty relations that can be defined from $A$ to $B$ is:

If $R$ is a relation on finite set $A$ having $n$ elements, then the number of relations on $A$ is

If $n\left(A\right)=2$ and $n\left(B\right)=3$, then write the total number of non empty relations that can be defined from $A$ to $B$

Let $A=\left\{1,2,3,4,5\right\}$ and let $B=\left\{1,2,3,4\right\}$. If the relation $R:A\to B$ is given by $\left(a,b\right)\in R$ if and only if $a+b$ is even, then $n\left(R\right)$ is equal to

Let $A=\left\{x:x^2<26\text{ and} x\text{ is a prime number}\right\} \text{and} B=\left\{y:\left|y\right|⩽1 \text{and} y \text{is a whole number}\right\},t\text{hen the number of relations from} B\text{ to} A \text{will be}$

If $\left(a,b\right)\in R$ such that $a^2+1=b$ where relation $R$ is subset of $A\times B,$ then the number of elements in range of relation $R$ $\left(\text{where} A=\left\{-1,0,1,2\right\} \text{and} B=\left\{1,2,3\right\}\right)$.

Let $A$ and $B$ be two smallest sets such that $A\cup \left\{1\right\}=\left\{1,2,3,4\right\}$ and $B\cup \left\{5\right\}=\left\{4,5,6,7,8\right\}.$ If $P=A-B$ and $Q=B-A$, then the number of relations from $P$ to $Q$ is

If $(a,b)\in R$ such that $a^2+1=b$ where relation $R$ is subset of $A\times B$, then number of elements in range of relation $R$ (where$A=\{-1,0,1,2\}\text{ and }B=\{1,2,3\}$.

Set $A$ has $5$ elements, the set $B$ has $4$ elements, the number of relations from a set $A$ to a set $B$ is 

A type of representation that represents a relation is

For the function $h\left(x\right)={10}^{1+x}$ the domain of real values of $x$ where $0\le x\le 9$, the range is

The range of the function $f\left(x\right)={\log }_{10}(1+x)$ for the domain of real values of $x$ when $0\le x$$\le 9$ is

The function $f\left(x\right)=2^x$ is