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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:

$\displaystyle A \cup B = \{ x \; \vert \; x \in A$    (inclusive) or $\displaystyle x \in B \}$    

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
\begin{figure}
\centering
\mbox{\epsfig{file=union.eps,height=3.5cm}}
\end{figure}

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

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

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:

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

Figure 7: The associative law for union of sets
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\centering
\mbox{\epsfig{file=UnionAssoc.eps,height=4cm}}
\end{figure}

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

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

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

$\displaystyle A \cup \emptyset$ $\displaystyle = A$    
$\displaystyle A\cup A$ $\displaystyle = A$    

This proposition can be proven with truth tables.

Proposition 3.2.17   For any two sets, the following holds:

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

Here too, use truth tables.
next up previous contents
Next: Distributivity Up: Operations with sets Previous: Intersection   Contents
root 2002-06-10