## Foundations of Mathematics## Set Theory |

## What are the basic set operations? |

There are several basic set operations, the most common being the intersection of sets, union of sets, and the complement of sets. The following lists these operations (note: the first two operations obey the associative and communtative laws, and together they obey the distributive law):

*Intersection*—The intersection of two sets is the set of elements common to the two sets. For example, the intersection of sets *A* and *B* is the set of elements common to both *A* ∩ B. This is usually written as *A* D B. Thus, if *A* = {1, 2, 3, 4} and *B* = {3, 4, 5}, then the intersection of *A* and *B* would be {3, 4}.

*Union*—The union of sets is the combining of members of the sets. For example, the union of two sets *A* and *B* is the set obtained by combining members of sets *A* and B. This is usually written as *A ∪ B*. Thus, if *A* = {1, 2, 3, 4} and *B* = {3, 4, 5}, then the union of *A* and *B* would be {1, 2, 3, 4, 5}.

*Complement or complementation*—When the set of all elements under consideration must be specified, it is called the universal set. And if the universal set is *U* = {1, 2, 3, 4, 5} and *A* = {1, 2, 3}, then the complement of A (or A’) is the set of all elements in the universal set that are not *A,* or {4, 5}. The intersection between a set and its complement is the empty or null set (∅); the union of a set and its complement is the universal set.