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An Introduction to Set Theory

A set is a collection of objects.  We group numbers into sets for various reasons, including one of the most common groupings: domains and ranges of functions.  In order to work with sets of numbers effectively, we need to understand a bit of the basic theory of sets.

There are often many ways to define a set.  The following three examples consist of the same set defined in two different ways.

a)  A = .
     This is a simple list of all members of set A.

b)  Set A can also be described as the set of all even integers larger than 0 and
     less than 20. 

c)   We can further describe Set A as {x | 2 ≤ x ≤ 18 and x is even.}  This is called
      set-builder notation and it uses a more formal description of the objects in the
      set.  This notation would be read, “The set of all values of x such that x is
      greater than or equal to 2 and less than or equal to 18 and x is even.”

A capital letter is often used to denote a set.

For the following examples: 

A =

B =

C = .

The objects in a set are called elements of the set.
The number 6 is an element of A.  This is written in symbols as
The number 5 is less than 20, but it is odd.  Therefore, 5 is not an element of A.  This is written in symbols as .

Set B is a subset of set A if and only if every element of B is also an element of A.  Even if A and B were identical sets, then B would still be considered to be a subset of A. 
Using our definitions of A, B, and C, above, set B is a subset of A.
This is written in symbols as .
The set C is not a subset of A because it contains elements not in A (namely, 13 and 15). 
This is written in symbols as .

Set B is a proper subset of set A if and only if every element in B is an element of A AND A contains at least one element that is not in B.  The set B is a proper subset of A, since 2, 4, 6, 8, 10 are elements of A and are not elements of B, and also every element of B is in A.
This is written in symbols as .

The union of sets A and B, written A U B, is the set of all elements contained in A or contained in B.      
A U B = {2, 4, 6, 8, 10, 12, 14, 16, 18}

The intersection of sets A and B, written A ∩ B, is the set of all elements that A and B have in common.           
A ∩ B = {12, 14, 16, 18}

The difference of sets A and B is defined to be the set of elements that are in A but not in B.
A - B = {2, 4, 6, 8, 10}

The empty set is the set without any elements.  It is written as

The Cartesian product of sets A and B is the set of all ordered pairs (a, b) where a is in set A and b is in set B.

 Let A =  and B =

Then

A Venn diagram is a diagram which illustrates the relationships between sets. The Venn diagram below represents the intersection of sets A and B (e.g., A U B). The shaded region represents the set of elements that belong to set A and to set B.