Congruence

Complement

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Complement generally refers to

Absolute complement

In general, let s be a set and a be a subset of S. the set composed of all elements not belonging to a in S is called the sum of a subset in S

absolutely

Complement

. In other branches of set theory and mathematics, there are two definitions of complement: relative complement and absolute complement.

definition

In other branches of set theory and mathematics, there are two definitions of complement: relative complement and absolute complement.

1. Relative complement

If a and B are sets, then the relative complement of a in B is such a set: its elements belong to B but not to a, B-A = {x | x ∈ B and X ∉ a}.

2、

Absolute complement

If there is a ⊆ u for a complete set u, then a is in U

relative complement

It's called a

Absolute complement

(or abbreviation)

Complement

）, writing ∁ < sub > U < / sub > a.

be careful

: Learning

repair

collection

First of all, we should understand the relativity of the complete set

1. A is a subset of u, namely a ⊆ U;

2. ∁ < sub > U < / sub > A represents a set, and ∁ < sub > U < / sub > a ⊆ u;

3. ∁ < sub > U < / sub > A is a set of all elements in u that do not belong to A. There are no common elements between ∁ < sub > U < / sub > A and a, and the elements in u are distributed in these two sets.

Complete set and complement

The complete set is a relative concept, which only contains all the elements involved in the problem,

Complement

Only relative to the corresponding complete set. If we study the problem in the range of integers, then

Z

When the problem is extended to real number set, then

R

For the complete set, the complement is only relative to this.

Correlation operation

Complement law and difference set

De Morgan's law

Morgan's law, also known as inversion law, can be simply described as follows: the complement of the intersection of two sets is equal to the union of their respective complements, and the complement of the union of two sets is equal to the intersection of their respective complements.

If sets a and B are two subsets of the complete set u, then the following relation holds

(1) ∁ < sub > U < / sub > (a ∩ b) = (∁ < sub > U < / sub > A) ∪ (∁ < sub > U < / sub > b), that is, the "complement of intersection" is equal to the "complement of union";

(2) ∁ < sub > U < / sub > (a ∪ b) = (∁ < sub > U < / sub > A) ∩ (∁ < sub > U < / sub > b), that is, the "complement of combination" is equal to the "intersection of complement".