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;
- *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.
Referenced By
Axiomatic Set Theory | Basic Set Theory | Complement (sets) | Formal set theory | List of basic discrete mathematics topics | List of mathematical logic topics | List of mathematical topics (S-U) | Naive Set Theory | Naïve set theory | Set-theoretic difference
|
