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:

Figure 6:
The union of two sets

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:

Figure 7:
The associative law for union of sets

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: