Question

Let U be a finite, nonempty set and let P(U) be the power set of U. That is, P(U) is the set of all subsets of U. Define the relation ~ on P(U) as follows: For A, B P(U), A B if and only if A B = That is, the ordered pair (A, B) is in the relation ~ if and only if A and Bare disjoint. Is the relation an equivalence relation on P(U)? If not, is it reflexive, symmetric, or transitive? Justify all conclusions.

Answer 1

The relation ~ on P(U) is not an equivalence relation as it is not reflexive. However, it is symmetric and transitive because sets that are disjoint from each other maintain this property in ordered pairings.

Step-by-step explanation:

When considering the power set P(U) of a set U, the relation ~ is defined such that for A, B in P(U), A ~ B if and only if A and B are disjoint. To be an equivalence relation, a relation must be reflexive, symmetric, and transitive. However, relation ~ is not reflexive because no set can be disjoint with itself, hence an ordered pair (A, A) cannot be in the relation. It is symmetric because if A is disjoint from B, then B is also disjoint from A. Lastly, it is transitive; if A is disjoint from B and B is disjoint from C, then A must be disjoint from C. Therefore, the relation is symmetric and transitive but not an equivalence relation.