Disjoint union

In the mathematical subfield of set theory, there are two slightly different uses of the term disjoint union.

  • Example 3.1 according to the first definition
  • Example 3.2 according to the second definition

Definition

The following distinction is exactly the difference between internal and external direct sum.

First variant: association of disjoint sets

A set is the disjoint union of a system of subsets, written

If the following two conditions are met:

  • If, that is, which are therefore pairwise disjoint;
  • , Ie is the union of all sets.

Second variant: Disjoint union of an arbitrary number

Are quantities for given, ie, the amount

The disjoint union of the sets. It is about an association, in which the amounts previously made ​​artificially disjoint.

Properties

  • For the widths applies.
  • The disjoint union is the categorical coproduct in the category of sets. This means that pictures correspond uniquely systems of pictures with.
  • If the quantities disjoint, then the canonical map is bijective.

Examples

For example according to the first definition

Disjoint union of and.

  • Both sets are disjoint
  • Is the disjoint union of the sets and ⇒
  • The quantities and thereby form a partition of the set

For example according to the second definition

Disjoint union of and.

  • Set theory
Mathematics
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