| 单词 | algebra of sets |
| 释义 | algebra of sets (i) A∪(B ∪ C)=(A ∪ B)∪C, A + (B + C)=(A + B) + C and A∩(B ∩ C)=(A ∩ B)∩C, the associative properties. (ii) A ∪ B = B ∪ A, A + B = B + A, and A ∩ B = B ∩ A, the commutative properties. (iii) A∪Ø = A, A + Ø = A and A∩Ø= Ø, where Ø is the empty set. (iv) A ∪ E = E and A ∩ E = A. (v) A ∪ A = A, A + A = Ø and A ∩ A = A. (vi) A∩(B ∪ C)=(A ∩ B)∪(A∩∩C), A + (B ∩ C)=(A + B)∩(A + C), and A∪(B ∩ C)=(A ∪ B)∩(A ∪ C), the distributive properties. (vii) A ∪ A′ = E, A + A′ = E, and A ∩ A′=Ø. (viii) E′=Ø and Ø′ = E. (ix) (A′)′ = A. (x) (A ∪ B)′ = A′∩B′ and (A ∩ B)′ = A′∪B′, De Morgan's laws. The application of these laws to subsets of E is known as the algebra of sets. Despite some similarities with the algebra of numbers, there are important and striking differences. If |E| = n then (℘(E), +, ∩) is isomorphic as a ring to (ℤ2n, +, ×). See Boolean algebra. |
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