Using indirect proof, solve this problem

Question

Show using indirect proof. Note that you will also need to use a case analysis here, once you set up the indirect proof. Don’t forget that even if you’re doing informal proofs, you can appeal to things like DeMorgan’s law.

∀x. x ∉ P(A ∪ B) → (x ⊈ A ∧ x ⊈ B)

(In other words: we want to show that if an element isn’t in the powerset of A ∪B, it is neither a subset of A nor a subset of B.)

Any help would be appreciated, thank you

Answer 1

Let x be any set.

Suppose ¬(x⊄A ∧x⊄B). Then x⊆A or x⊆B, by DeMorgan's Law.

If x⊆A, then x⊆AUB, since A⊆AUB. Thus, x∈Ρ(AUB), since x⊆AUB.

If x⊆B, then x⊆AUB, since B⊆AUB. Thus, x∈Ρ(AUB), since x⊆AUB.

In either case, x∈Ρ(AUB), which completes the indirect proof.