Definition: Union

Definition: Let A and B be sets. Then the union of A and B, often denoted by $A \cup B$ , is the set whose elements are precisely those that are either in A or in B. Written in the notation of lesson 16, we have that $A\cup B =\{p|p\in A \mathrm{\ or\ } p\in B\}$ .//

It is important to remember that $A \cup B$ is itself a set just like any other set. Thus, we’ve taken the data of the sets A and B and created a new set, $A\cup B$ . We note that the union of a set with any of its subsets (including and especially the empty set) is just the original set, because the subset “brings in nothing new”. For example, $\{1, 2, 3, 4, 5\} \cup \{2, 4, 5\}=\{1, 2, 3, 4, 5\}$ since a set only sees its distinct elements. We also note that a set A is always contained in the union of A with anything, simply because the union of A with anything certainly contains (at least) all of A’s elements.