The intersection of two sets and is the set that contains all the elements such that and . The intersection of and is written as . The following Venn diagram illustrates the intersection of two sets and :
Example. If and then .
We can also define the intersection of an arbitrary number of sets. If is a family of sets, we define the intersection of all them, denoted , as the set consisting of those elements belonging to every set :
A set intersects, or meets, a set if is non-empty.
Remark. What is when ? In other words, what is the intersection of an empty family of sets? First note that if , then
This leads the conclusion that the intersection of an empty family of sets should be as large as possible. How large should it be? In addition, is this intersection a set? The answer depends on what versions of set theory we are working in. Some theories (for example, von Neumann-Gödel-Bernays) say this is the class of all sets, while others do not define this notion at all. However, if there is a fixed set in advance such that each , then it is sometimes a matter of convenience to define the intersection of an empty family of to be .
|Date of creation||2013-03-22 12:14:52|
|Last modified on||2013-03-22 12:14:52|
|Last modified by||CWoo (3771)|