The Language of Sets
Set theory
Chapter 2 Sec. 1
Key Words
 What is a set?
 Collection of objects.
 Use of capital letters to name sets.
 What is a element/member?
 Individual objects in a set.
 Use of lowercase letters to denote
elements in a set.
How to represent a set?
Consider the set of seasons of the
year to be the set S.
S = {Spring, Summer, Fall,
Winter}.
Set Builder Notation
Set builder notation is to represent
the set, if the elements of a set all
share some common characteristics
that are satisfied by no other
objects.
Examples
C = {x:x is a carnivorous animal}
 is is equivalent to =
{ is the set
x “of all x”
: is such that
 We can use set builder notation for the
solution we will have to write.
 C={lion, tiger, panther}
Write an alternative method.
 B={y:y is a color of the state of New
Mexico flag.}
 B={yellow and red}
 A={a:a is counting number less than 20
and is evenly divisible by 3.}
A={3,6,9,12,15,18}
Well defined
A set is well defined if we are
able to tell whether any
particular object is an element of
the set.
Example
 Here is two examples, which sets are
well defined?
 A) M = {x:x is a mountain over 10,000
ft high}
Well defined
 B) S={s:s is a scary movie}
Not well defined
How about this problem?
 M = {m:m is in your math class and is
also a star on the Sopranos.}
 This set has no elements.
Empty set or Null set
The set that contains no elements is
called the empty set. This set is
labeled by the symbol Ø. Another
notation for the empty set is {}.
Universal set
Is the set of all elements under
consideration in a given discussion.
We often denote the universal set
by the capital U.
Example
 Consider U = {0, 1, 2, 3, …9, 10}
 U = {x:x is a male consumer living in
the United States.}
Elemental symbol
 We will use the symbol
to stand
for the phrase is an element of.
How is it used?
Example
 The notation
4  A is expresses
that 4 is an element of the set A.
 The notation
4  A is expresses
that 4 is not an element of the set A.
Use either
 A) 3


{2, 4, 3, 5}

 B) {4} {2, 3, 4, 5}


 4 {x:x is an odd counting number}
Cardinal Number
 The number of elements in set A and
denoted by n(A). A set is finite if its
cardinal number is a whole number. An
infinite set is one that is not finite.
Example problems
 State whether the set is finite or
infinite. If it is finite, state its cardinal
number using n(A) notation.
 P = {x:x is a planet in our solar
system}.
 N ={1,2, 3}