Union (mathematics)

Definition

In the Set theory, the union or meeting of two Ensemble S has and B is the unit which contains all the elements which belong to has or belong to B . One notes the union of has and B has ∪ B. In notation symbolic system, it is:

\ forall X, X \ in has \ cup B \ Leftrightarrow ((X \ in A) \ lor (X \ in B))

For example the union of the units has = {1,2,3} and B = {2,3,4} is the unit {1,2,3,4}.

In Boolean Algebra, the union is associated with the logical Operator ou inclusif.

One generalizes this concept with a family of units $(A_i) _ {I \ in I}$ . The meeting or union of the members units of this family is the whole of the elements $x$ for which there exists a $i \ in I$ such as $x \ in A_i$ . It then is noted $\ bigcup_ {I \ in I} A_i$ .

Algebraic properties

the union is associative, i.e for units has , B and C unspecified, one a:

( has ∪ B ) ∪ C = has ∪ ( B ∪ C )

the union is commutative, i.e for units has and B unspecified, one a:

has ∪ B = B ∪ has

the Intersection is distributive on the union, i.e for units has , B and C unspecified, one a:

has ∩ ( B ∪ C ) = ( has ∩ B ) ∪ ( has ∩ C )

Fiu-vro: Hulkõ kogo Zh-classical: 並集

Random links: