Question

Given a non empty set X, 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 if A ⊂ B. Is R an equivalence relation on P(X)? Justify you answer:

Answer 1

$\textbf{\underline{conditions of equivalence:}}\\\textbf{1. relation should be reflexive}\\\textbf{2. relation should be symmetric}\\\textbf{3. relation should be transitive.}$

now, We know that every set is a subset of itself.
so, ARA for all A ϵ P(X).
therefore,  R is reflexive.
This cannot be implied to B ⊂ A [ B is subset of A]
So, if A = {a, b} and B = {a, b, c}, then it cannot be implied that B is related to A.
therefore,  R is not symmetric.
So, if ARB and BRC, then A ⊂ B and B ⊂ C.
⇒ A ⊂ C
therefore, R is transitive.
Therefore, R is not an equivalence relation since it is not symmetric.