Question

Let ${\rho }_1$ and ${\rho }_2$ be two equivalence relations defined on a non-void set S. Then

• A. both ${\rho }_1\cap {\rho }_2$ and ${\rho }_1\cup {\rho }_2$ are equivalence relations.
• B. ${\rho }_1\cap {\rho }_2$ is equivalence relation but ${\rho }_1\cup {\rho }_2$ is not so.
• C. ${\rho }_1\cup {\rho }_2$ is equivalence relation but ${\rho }_1\cap {\rho }_2$ is not so.
• D. neither ${\rho }_1\cap {\rho }_2$ nor ${\rho }_1\cup {\rho }_2$ is equivalence relation

Answer 1

The correct option is B ${\rho }_1\cap {\rho }_2$ is equivalence relation but ${\rho }_1\cup {\rho }_2$ is not so.

Given ${\rho }_1,{\rho }_2$ are equivalence relations on $S$.
$\Rightarrow$ ${\rho }_1,{\rho }_2$ are reflexive, symmetric and transitive.

Reflexive:
Let $x\in S$
$\Rightarrow (x,x)\in {\rho }_1$ and $(x,x)\in {\rho }_2$
$\Rightarrow (x,x)\in {\rho }_1\cap {\rho }_2$
$\Rightarrow {\rho }_1\cap {\rho }_2$ is reflexive.

Symmetric:
Let $(x,y)\in {\rho }_1\cap {\rho }_2$
We have to show $(y,x)\in {\rho }_1\cap {\rho }_2$
$(x,y)\in {\rho }_1\cap {\rho }_2$
$\Rightarrow (x,y)\in {\rho }_1$ and $(x,y)\in {\rho }_2$
$\Rightarrow (y,x)\in {\rho }_1$ and $(y,x)\in {\rho }_2$
$\Rightarrow (y,x)\in {\rho }_1\cap {\rho }_2$
$\Rightarrow {\rho }_1\cap {\rho }_2$ is symmetric.

Transitive:
Let $(x,y),(y,z)\in {\rho }_1\cap {\rho }_2$
$\Rightarrow (x,y),(y,z)\in {\rho }_1$ and $(x,y),(y,z)\in {\rho }_2$
$\Rightarrow (x,z)\in {\rho }_1$ and $(x,z)\in {\rho }_2$
$\Rightarrow (x,z)\in {\rho }_1\cap {\rho }_2$
$\Rightarrow {\rho }_1\cap {\rho }_2$ is transitive.

Therefore, ${\rho }_1\cap {\rho }_2$ is equivalence relation.

${\rho }_1\cup {\rho }_2$ is always reflexive and symmetric but not transitive.
e.g. Let $S=\{1,2,3\}$
${\rho }_1=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\}$
${\rho }_2=\left\{\right(1,1),(2,2),(3,3),(2,3),(3,2\left)\right\}$

${\rho }_1,{\rho }_2$ is equivalence relation.
But ${\rho }_1\cup {\rho }_2$ is not transitive as $(1,2),(2,3)\in {\rho }_1\cup {\rho }_2$ but $(1,3)\notin {\rho }_1\cup {\rho }_2$