Wikipedia: Simple theorems in set theory

Simple theorems in set theory
From Wikipedia, the free encyclopedia.

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;

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:

A'' = A;

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.