Question

Given a non empty set X, consider $P\left(X\right)$ which is set of all subsets of $X$. Define the relation $R$ is $P\left(X\right)$ as follows:
For subsets $A,B$ in $P\left(X\right),ARB$ if and only if $A\subset B$. Is R an equivalence relation on $P\left(X\right)$? Justify your answer

Answer 1

$LetX=\{1,2,3\}$

$P\left(X\right)$=Power set of $X$=Set of all subsets of $X$.

=$\{\varphi ,\{1\left\}\right\},\left\{2\right\},\left\{3\right\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\left\}Since\right\{1\}\subset \{1,2\left\}\right\{1\left\}R\right\{1,2\}$

$\because ofABCD,allelementsofAareinB$

$ARB$ means $A\subset B$

here,relation is $R=\left\{\right(A,B):AandBaresets,ACB\}$

Since every set is a subset of itself.

$ACA\therefore (A,A)ϵR,$R is reflexive.

To check whether symmetric or not,

If$(A,B)ϵ,then(B,A)ϵR.$

$If(A,B)ϵR,A\subset BbutB\subset Aisnottrue.eg:-LetA=\left\{1\right\}andB=\{1,2\}$

As all elements of $A$ are in $BA\subset B$.

But all elements of $B$ are not in $A$

$\therefore B\subset A$ is not true.

$\therefore R$ is not symmetric.

Since $(A,B)ϵRand(B,C)ϵRofA\subset BandB\subset C$

then $A\subset B$

$\Rightarrow (A,C)ϵRSo,If(A,B)ϵRand(B,C)ϵR,then(A,C)ϵR$

$\therefore R$ is transitive.

hence $R$ is reflexive and transitive but not symmetric.

hence, $R$ is not an equivalence relation since it is not symmetric.