Definition: Let A and B be sets. Then the union of A and B, often denoted by , 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 .//
It is important to remember that 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, . 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, 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.
For more on unions, check out lesson 16.