Sets:Operations

From Department of Mathematics at UTSA
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Definitions

The two main set operations that we deal with are union and intersection. The union of two sets and is defined as or . For example:

  • The union of and is
  • The union of the even integers and odd integers is .
  • The union of the set of rational numbers and the set of irrational numbers is .
  • , and .
  • For sets and such that , , since all elements of are already in if .

The intersection of and is defined as and ; that is, the intersection of and is the set of all elements shared by the two sets. Sets and are "disjoint" if .

  • The intersection of and is .
  • The intersection of the even integers and odd integers is the empty set, since no element between these two sets is shared (an integer cannot be both even and odd).
  • , and .
  • For sets and such that , .

There are a few other common set operations. The set difference of and is defined as . We read (also sometimes denoted as ) as " without ". Note that this operation is not commutative; that is, does not equal in most cases. Example: if and , then and .

Another set operation is the Cartesian product (or the product of two sets). The Cartesian product of two sets and is defined as and , where is an ordered pair. For example, if and , then . If and are both finite sets, then the number of elements in is the product of the number of elements in set and set (for example, if has 3 elements and has 2, then has 6).

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