Definition: Let A and B be sets. Then the intersection of A and B, often denoted by , is the set whose elements are precisely those that are in A and 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 intersection of a set A with anything else is always contained in A. This is because intersecting A with anything only limits what can be in the intersection, because it puts an extra restriction on what the elements can be (namely, that they need to be in A and in whatever set we’re intersecting A with). For example, . It is also possible to have an empty intersection even when the two sets that we’re intersecting are not empty. This just means that the sets have no elements in common. This is still in accord with what was said above—that the intersection of A with anything is a subset of A—because the empty set is a subset of A!
For more on intersections, check out lesson 16.