Example Proof in Set Theory

Example Proof in Set Theory

1. Proposition: "∀A, B sets, (A ∩ B) ∪ (A\B) = A".

   • Show (A ∩ B) ∪ (A\B) ⊆ A
     • ∀x ∈ (A ∩ B) ∪ (A\B), x ∈ (A ∩ B) or x ∈ A\B
       • If x ∈ (A∩B), then clearly x ∈ A as A∩B ⊆ by definition.
         • If x ∈ A\B, then by definition, x ∈ A and x !∈ B, so definitely x ∈ A.
           • In both cases, x ∈ A as needed.
   • Show A ⊆ (A ∩ B) ∪ (A\B)
     • Either...
       • x ∈ B
         • ... then x ∈ A and x ∈ B, so x ∈ A ∩ B
           • ... in both cases
             • x ∈ (A ∩ B) or x ∩ (A\B)
               • so x ∈ (A ∩ B) ∪ (A\B), as needed
       • x !∈ B
         • x ∈ A and x !∈ B, so x ∈ A\B
   • Q.E.D.