Generally speaking, let S be a set, A is a subset of S, and the set composed of all elements that do not belong to A in S is called the absolute complement of subset A in S. In other branches of set theory and mathematics, there are two definitions of complement set: relative complement set and absolute complement set.
Given a complete set u, there is one? U, then the relative complement set of A in U is called the absolute complement set of A (or simply the complement set). UA .
Note: To learn the concept of complement, we must first understand the relativity of complete works and the symbols of complement. UA has three meanings:
1, a is a subset of u, that is, a? u；
2、? UA stands for a set, and? UA？ u；
3、? UA is a set of all elements in U that do not belong to A. UA and A have no common elements, and the elements in U are distributed in these two sets.
Extended data:
A complete set is a relative concept, which only contains all the elements involved in the studied problem, and a complementary set is only relative to the corresponding complete set. For example, when we study the problem in the integer range, Z is a complete set, and when the problem is extended to the real number set, R is a complete set, and the complementary set is only relative to this.
If sets A and B are two subsets of complete set U, the following relationship holds:
( 1)? U(A∩B)=(？ UA)∩(？ UB), that is, "the complement of the cross" is equal to "the combination of complements";
(2)? U(A∪B)=(？ UA)∩(？ UB), that is, "the complement of merger" equals "the turning point of complement".
Certainty: given a set, any element, whether it belongs to the set or not, must be one of them, and no ambiguity is allowed? .
Reciprocity: Any two elements in a set are considered different, that is, each element can only appear once. Sometimes it is necessary to describe the situation where the same element appears many times. You can use multiset, where elements are allowed to appear multiple times.
Disorder: In a set, each element has the same state and the elements are out of order. You can define an order relation on the set. After defining the order relation, you can sort the elements according to the order relation. But as far as the characteristics of the set itself are concerned, there is no necessary order between elements.
If the elements of two sets S and T are exactly the same, then the two sets S and T are equal, which is marked as S = T. Obviously, there is the following relationship: the symbol is called if and only if, which means that the proposition on the left and the proposition on the right contain each other, that is, the two propositions are equivalent.
Operating rules:
Exchange law: a ∩ b = b ∩ a; A∪B=B∪A
Law of constraint: a ∪ (b ∪ c) = (a ∪ b) ∪ c; A∩(B∩C)=(A∩B)∩C
The law of distribution duality: a ∩ (b ∪ c) = (a ∪ b) ∩ (a ∪ c); A ∪( B∪C)=(A∪B)∪( A∪C)
Duality law: (a ∪ b) c = a c ∪ b c; (A∩B)^C=A^C∪B^C
Identity: A∨? = A； A∩U=A
The law of seeking complement: a ∪ a' = u; A∩A'=？
Law of involution: A''=A
Idempotent law: a ∪ a = a; A∩A=A
Zero uniformity: a ∪ u = u; A∩？ =?
Law of absorption: a ∨ (a ∩ b) = a; A∩(A∪B)=A
Inverse Law (De Morgan Law): (a ∪ b)' = a' ∪ b'; (A∩B)'=A'∪B ' .
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