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Union

Definition 3.2.13   Let and be two sets. The union of and is the set of all the elements that belong to or to (or to both). We denote:

 (inclusive) or

Note that and are subsets of . The union of two sets can be repesented using Venn diagrams:

Proposition 3.2.14 (the commutative law)   Let and be two sets. Then the following equality holds:

The commutative law for the union of sets can be generalized to the union of any (finite) number of sets; the proof is done by induction (v.i. Definition def Peano).

Proposition 3.2.15 (the associative law)   Let ,and be three sets. Then the following equality holds:

The associative law for the union of sets can be generalized to the union of any (finite) number of sets; the proof is done by induction (v.i. Definition def Peano). An extension of the associative law for union is as follows. Let and be two sets of indices; if and are two families of sets, then

Proposition 3.2.16   Let be any set. Then we have:

This proposition can be proven with truth tables.

Proposition 3.2.17   For any two sets, the following holds:

Here too, use truth tables.

Next: Distributivity Up: Operations with sets Previous: Intersection   Contents
root 2002-06-10