6.2 Properties of Sets

Theorem 6.2.1: Some Subset Relations

• For all sets, (a) A ∩ B ⊆ A and (b) A ∩ B ⊆ B
• For all sets, (a) A ⊆ A ∪ B and (b) B ⊆ A ∪ B
• For all sets, if A ⊆ B and B ⊆ C, then A ⊆ C

Theorem 6.2.2: Set Identities

• Commutative Laws: (a) A ∪ B = B ∪ A and (b) A ∩ B = B ∩ A
• Associative Laws: (a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and (b) (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Distributive Laws: (a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and (b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
• Identity Laws: (a) A ∪ ∅ = A and (b) A ∩ U = A
• Complement Laws: (a) A ∪ A^c = U and (b) A ∩ A^c = ∅
• Double Complement Law: (A^c)^c = A
• Idempotent Laws: (a) A ∪ A = A and (b) A ∩ A = A
• Universal Bound Laws: (a) A ∪ U = U and (b) A ∩ ∅ = ∅
• DeMorgan's Laws: (a) (A ∪ B)^c = A^c ∩ B^c and (b) (A ∩ B)^c = A^c ∪ B^c
• Absorption Laws: (a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A
• Complements: (a) U^c = ∅ and (b) ∅^c = U
• Set Difference Law: A - B = A ∩ B^c

Theorem 6.2.3: Intersection and Union with a Subset

For any sets A and B, if A ⊆ B, then (a) A ∩ B = A and (b) A ∪ B = B

Theorem 6.2.4: A Set with No Elements is a Subset of Every Set

If E is a set with no elements and A is any set, then E ⊆ A

Basic Method for Proving That Sets Are Equal

Let sets X and Y be given. To prove that X = Y

• Prove that X ⊆ Y
• Prove that Y ⊆ X

Element Method for Proving a Set Equals the Empty Set

To prove that a set X is equal to the empty set ∅, prove that X has no elements. To do this, suppose X has an element and derive a contradiction.

In Class Exercises

• 5 - Prove that for all sets A and B, (B - A) = B ∩ A^c
• 15 - Prove that for every set A, A ∪ ∅ = A
• 30 - Prove that for every subset A of a universal set U, A ∩ A^c = ∅