Question

If a relation $R$ is defined on $A=\{1,2,3,4\}$, then which of the following is/are universal relation?

• A. $R=\left\{\right(a,b):a+b<10:a,b\in A\}$
• B. $R=\left\{\right(a,b):\text{value of}a-b\text{is an integer}:a,b\in A\}$
• C. $R=\left\{\right(a,b):\text{value of}\frac{a}{b}\text{is an integer}:a,b\in A\}$
• D. $R=\left\{\right(a,b):ab\text{is a real number}:a,b\in A\}$

Answer 1

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

The correct options are
A $R=\left\{\right(a,b):a+b<10:a,b\in A\}$
B $R=\left\{\right(a,b):\text{value of}a-b\text{is an integer}:a,b\in A\}$
D $R=\left\{\right(a,b):ab\text{is a real number}:a,b\in A\}$

$R=\left\{\right(a,b):a+b<10:a,b\in A\}$
Here,
$a+b\le 8\therefore a+b<10$
It is a universal relation.

$R=\left\{\right(a,b):\text{value of}a-b\text{is an integer}:a,b\in A\}$
Here, $a,b$ are integers
$\therefore a-b$ will be an integer (always)
It is a universal relation.

$R=\left\{\right(a,b):\text{value of}\frac{a}{b}\text{is an integer}:a,b\in A\}$
Here, $R=\left\{\right(1,1),(2,1),(3,1),(4,1),(2,2),(3,3),(4,2),(4,4\left)\right\}$
It is not a universal relation.

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