The Language of SetsBasic Terminology

 

  Definition.  A set is a well-defined collection of objects or ideas.

 

What does it mean for a set to be well-defined?

            A set is well-defined, if there is no ambiguity as to whether or not an object belongs to it.

 

        Examples.  Which of the following are sets?

1.    The collection of good students
        The collection is not a set since the word “good” is ambiguous and the phrase is not well-defined.

2.    The collection of students at MSUM with a grade point average above 3.0.
         It is fairly clear which students would belong to this collection, hence, it is a set since it is well-defined.

3.    The collection of young people that are residents of Minnesota
         The collection is not a well-defined set since “young” is an ambiguous term.

4.    The collection of residents of Minnesota between the ages of 18 and 25 years old
           It is fairly clear who would belong to this collection, hence, it is a well-defined set.

 

We have three ways of describing sets:

1.   by name or verbal description of the elements of a set,

2.   by roster (list) form by listing the elements separated by commas and using braces to enclose the list, or

3.   by set-builder notation that uses a variable and a rule to describe the elements of a set.

 

      Examples.

        Let A represent the collection of states that border Minnesota.

      A = {North Dakota, South Dakota, Iowa, Wisconsin, Michigan}

      A = {x : x is a state bordering Minnesota}
       This is read as “A is the set of all x such that x is a state bordering Minnesota”.

 

      Let B represent the collection of counting numbers.

      B = {1, 2, 3, 4, … }

      B = {x : x is a counting number}

 

  Definition.  An object or idea in a set is called an element (member) of the set. The symbol  is used to denote that an element is a member of a set and  is used to denote that an object is not a member of a set.

 

      Examples. For the sets A and B used in the previous example, we have

      Iowa  A,   Alabama  A,  8  B,  and  0  B.

 

Definition. The null set (empty set) is a set that has no members. The symbol     is used to represent the null set (empty set).

 

      Example.

  = The collection of people attending MSUM who are 200 years old

  = { }

  = {x : x is a person attending MSUM who is 200 years old.}

 

Note that { } does not symbolize the empty set; it represents a collection of empty sets.

 

Definition. Two sets are equal, if they have exactly the same elements.

 

      Examples.

      Consider the sets A = {a, b, c}, B = {b, c, a}, C = {a, a, c, b, c}, D = {a, b, d},
then A = B = C ≠ D.  The sets A, B, and C are all equal since they each have the elements a, b, and c, and no other elements. The set D is not equal to the other three sets since c A  but  c  D.

 

      Note that the order of the elements does not matter and an element does not need to be listed more than once.

 

     

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