Question

Consider the following statements,
1. Identity. relation in a finite set $A$ is the greatest relation in $A$ .
2. The universal relation in a set containing at least $2$ elements is not anti-symmetric.
3. The union and intersection of two symmetric relations are also symmetric relations.
Which of these is / are correct?

• A. Only $1$
• B. $2$ and $3$
• C. $1$ and $3$
• D. All of these

Answer 1

Answer: (B)

$1.$ Universal relation is the largest relation and identity relation in a finite set $A$ is a subset of universal relation. Therefore, identity relation on a finite set $A$ is not the greatest relation.
$2.$ Consider the set $A=\left\{\right.1,2\left.\right\}$
$\Rightarrow R\left\{\right.\left(\right.1,1\left.\right),\left(\right.1,2\left.\right),\left(\right.2,1\left.\right),\left(\right.2,2\left.\right)\left.\right\}$
$\Rightarrow R$ is a symmetric relation
$\Rightarrow R$ is not anti symmetric
$3.$ By definition, union and intersection of $2$ symmetric relations are also symmetric relation.
So, only $2$ and $3$ are correct.