Simple theorems in set theory
We list without proof several simple properties of these operations. These properties can be visualized with Venn diagrams.
PROPOSITION 1: For any sets A, B, and C:
- A ∪ A = A;
- A \\ A = {};
- A ∩ B = B ∩ A;
- A ∪ B = B ∪ A;
- (A ∩ B) ∩ C = A ∩ (B ∩ C);
- (A ∪ B) ∪ C = A ∪ (B ∪ C);
- C \\ (A ∩ B) = (C \\ A) ∪ (C \\ B);
- C \\ (A ∪ B) = (C \\ A) ∩ (C \\ B);
- C \\ (B \\ A) = (A ∩ C) ∪ (C \\ B);
- (B \\ A) ∩ C = (B ∩ C) \\ A = B ∩ (C \\ A);
- (B \\ A) ∪ C = (B ∪ C) \\ (A \\ C);
- A ⊆ B if and only if A ∩ B = A;
- A ⊆ B if and only if A ∪ B = B;
- A ⊆ B if and only if A \\ B = {};
- A ∩ B = {} if and only if B \\ A = B;
- A ∩ B ⊆ A ⊆ A ∪ B;
- A ∩ {} = {};
- A ∪ {} = A;
- {} \\ A = {};
- A \\ {} = A.
PROPOSITION 2: For any universal set
U and subsets
A,
B, and
C of
U:
- B \\ A = A' ∩ B;
- (B \\ A)' = A ∪ B';
- A ⊆ B if and only if B' ⊆ A';
- A ∩ U = A;
- A ∪ U = U;
- U \\ A = A';
- A \\ U = {}.
PROPOSITION 3 (
distributive laws): For any sets
A,
B, and
C:
- (a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);
- (b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
The above propositions show that the power set
P(
U) is a
Boolean lattice.