Given a non empty set & consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: 

For subsets A, B in P(X), ARB if and only it A $\subset$B.  Then R is not

$\text{ Since every set is a subset of itself, }ARA\text{ for all }A\in P\left(X\right)$

Therefore, .R is reflexive. 

$\begin{array}{l}\text{ Let }ARB\Rightarrow A\subset B\text{ . }\\ \text{ This cannot be implied to }B\subset A\text{ . }\end{array}$

$\text{ For instant, if }A=\{1,2\}\text{ and }B=\{1,2,3\},\text{ then it cannot be }$

implied that B is related to A. 

Therefore, R is not symmetric. 

$\text{ Further if }ARB\text{ and }BRC\text{ , then }A\subset B\text{ and }B\subset C\text{ . }$

$\begin{array}{r}\Rightarrow A\subset C\\ \Rightarrow ARC\end{array}$

Therefore, R is transitive. Hence, R is not an equivalence relation since it is not symmetric