Question

Let $A$ be the set of first $10$ natural numbers. If a relation $R$ is defined on $A$, then the correct options(s) is/are

• A. $R=\left\{\right(a,b):a+b>20,a,b\in A\}$ is a void relation
• B. $R=\left\{\right(a,b):\frac{a}{b}\text{is a real number},a,b\in A\}$ is a universal relation.
• C. $R=\left\{\right(a,b):ab=1,a,b\in A\}$ is an identity relation.
• D. If $B=\{2,5,10\}$, then the number of common elements between $A\times B$ and $B\times A$ is $9$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $R=\left\{\right(a,b):a+b>20,a,b\in A\}$ is a void relation
B $R=\left\{\right(a,b):\frac{a}{b}\text{is a real number},a,b\in A\}$ is a universal relation.
C $R=\left\{\right(a,b):ab=1,a,b\in A\}$ is an identity relation.
D If $B=\{2,5,10\}$, then the number of common elements between $A\times B$ and $B\times A$ is $9$

$A=\{1,2,3,4,5,6,7,8,9,10\}$

$R=\left\{\right(a,b):a+b>20,a,b\in A\}$
As $a,b\in A$
$a+b\le 20$
$\therefore a+b$ can never be greater than $20$
It is a void relation.

$R=\left\{\right(a,b):\frac{a}{b}\text{is a real number},a,b\in A\}$ is a universal relation.
Since $a,b$ are real numbers
$\therefore \frac{a}{b}$ will be real number.
It is a universal relation.

$R=\left\{\right(a,b):ab=1,a,b\in A\}$
Here, $R=\left\{\right(1,1\left)\right\}$
It is an identity relation.

Here, the number of common elements between $A$ and $B$ is $3$.
$\therefore$ The number of common elements between $A\times B$ and $B\times A$ is $3^2=9$