Union

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

$$A \cup B = \{ x \; \vert \; x \in A x \in B \}$$

Note that $A$ and $B$ are subsets of $A \cup B$. 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 $A$ and $B$ be two sets. Then the following equality holds:

$$A \cup B = B \cup A.$$

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 $A$,$B$and $C$ be three sets. Then the following equality holds:

$$A \cup (B \cup C )= (A \cup B ) \cup C.$$

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 $I$ and $J$ be two sets of indices; if $\{ A_i , \; i \in I \}$ and $\{ b_j, \; j \in J \}$ are two families of sets, then

$$\left( \underset{i \in I}{\bigcup} A_i \right) \cup \left( \under... ...derset{\begin{matrix}i \in I \\ j \in J \end{matrix}} \bigcup (A_i \cup B_j).$$

Proposition 3.2.16   Let $A$ be any set. Then we have:

$$A \cup \emptyset = A$$

$$A\cup A = A$$

This proposition can be proven with truth tables.

Proposition 3.2.17   For any two sets, the following holds:

$$A \cup B = A \Longleftrightarrow B \subset A.$$

Here too, use truth tables.