Simple Theorems In The Algebra Of Sets

• 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.

• 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 = {}.

(a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);

(b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

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