Question

Let X={1,2,3}X={1,2,3}. Let P(X)P(X) be the set of all subsets of XX (i.e. Power set of XX). Let RR be a relation defined on P(X)P(X) by the following. For all sets AA and BB in P(X)P(X), ARBARB iff |A|=|B||A|=|B|. Is RR an equivalence relation?

Answer 1

Yes

Explanation:

For the equation to be equivalence then it must be

Reflexive, Symmetric and transitive

since |A|=|B|;

Every element in A is in B then R is reflexive

If f(1) = g(1), then g(1) = f(1), so R is symmetric.

If f(1) = g(1) and g(1) = h(1), then f(1) = h(1), so R is transitive.

R is reflexive, symmetric, and transitive,

thus R is an equivalence relation.