Complement (set theory)

In set theory and other areas of mathematics, two different complements are defined: the relative complement and the absolute complement.

• 2.1 Definition
• 2.2 Example
• 2.3 Properties

Relative complement

Definition

Are and quantities and is a subset of, then set-theoretical complement or set-theoretic difference of the quantity is the relative complement, called, precisely the elements of which are not included. The formal definition of the relative complement

And you say " B without A". The complement is different from the normal levels of subtraction only in the subset relationship must exist between the observed quantities. Relative states because the complement you can not specify an amount, without knowing the context. However, if the amount is fixed, so you can instead of " the relative complement of A in B" and simply " the complement of A" call.

Examples

• For ( real numbers ) and ( rational number ), the amount of the irrational numbers.

Properties

The following are some properties of relative complements in connection with the set-theoretic operations union and intersection are listed. Be, and quantities, then the following identities hold:

Absolute complement

Definition

Is a universe defined as the relative complement of in also absolute complement ( or simply complement) for each amount referred to and quoted as ( sometimes referred to as, or as, or if it is found ), so it is:

Example

Is the universe, for example, the set of natural numbers, the ( absolute ) complement of the set of even numbers, the set of odd numbers.

Properties

The following are some properties of absolute complements in connection with the set-theoretic operations union and intersection are listed. Let and be subsets of the universe, then the following identities hold:

De Morgan's rules:

Komplementgesetze:

• If, as is

Involution:

Relationships between relative and absolute complements:

The first two Komplementgesetze that if a non-empty subset of is, is a partition of.