6.3 Disproofs and Algebraic Proofs
Key Ideas
- To disprove an alleged set property, find a counterexample
- To prove that if set X has n elements, then P(X) has 2n
elements, use an induction proof
- Algebraic proofs can be used to derive set properties
Algebraic Proof Example: (A ∪ B) - C = (A - C) ∪︀ (B - C)
- Let A, B, C be any sets
- (A ∪ B) - C (starting point)
- = (A ∪ B) ∩ Cc (by set difference law)
- = Cc ∩ (A ∪ B) (by commutative law)
- = (Cc ∩ A) ∪ (Cc ∩ B) (by distributive law)
- = (A ∩ Cc) ∪ (B ∩ Cc) (by commutative law)
- = (A - C) ∪ (B - C) (by set difference law)
In Class Exercises
- 3 - Find a counterexample to show that the following statement is
false for all sets A, B and C. If A ⊄ B and B ⊄ C
then A ⊄ C.
- 18 - Disprove that for all sets A and B,
P(A ∪ B) ⊆ P(A) ∪︀ P(B)
- 31 - Construct an algebraic proof to show that for all sets
A and B, A ∪ (B - A) = A ∪ B