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 ), A R B if and only if A ⊂ B . Is R an equivalence relation on P( X )? Justify you answer:

Answer 1

It is given that P( X ) is the set of all subsets of X and the subsets A, B are defined in P( X ) such that ARB if and only if A⊂B.

As every set is a subset of itself, so ARA is applicable for all A∈P( X ). So, the relation is reflexive.

Consider a relation ARB such that A is the subset of B, that is A⊂B. But the inverse does not imply that B⊂A. Thus, the relation is not symmetric.

Consider the two relations ARB and BRC, such that A⊂B and B⊂C. So, correspondingly A⊂C and ARC, so the relation is transitive.

Thus, R is not an equivalence relation as the relation is not symmetric.