Lesson 1.1: Sets
A set is a collection of numbers or objects. We use the { } brackets (I often refer to these as “curly braces”)
to denote a set, and each element of the set is separated by a comma.
e.g. {2, 3, 5, 7, 11}
e.g. {a, e, i, o, u}
There are several important number sets that you need to be familiar with throughout this course.
: set of natural numbers
{0, 1, 2, 3, 4, . . .}
: set of integers
{. . ., -3, -2, -1, 0, 1, 2, 3, . . .}
: set of rational numbers
{
2
4
,  , 5, . . .}
3
7
: set of real numbers
{everything!}
A number set with a superscript of “+” includes only the
e.g.
+
members of the number set.
: set of positive natural numbers
A number set with a superscript of “*” includes only the
e.g.
members of the number set.
*: set of non-zero integers
A Venn diagram is used to represent sets of numbers or objects.
e.g. Draw a Venn diagram showing the relationships between the number sets
From this Venn diagram, we can see that
. The symbol  is the notation for
is a subset of
, which is a subset of
,
,
, and
, which is a subset of
. In other words, P  Q (“P is a subset of
Q”) means that every element in set P is also an element of set Q .
Hence, we can notate the relationships between the number sets
,
,
, and
in the following way:
.
The symbol  is the notation for
or
Similarly, the symbol  is the notation for
.
or
.
e.g. If M  {2, 3, 5, 7, 11} and N  {3, 6, 9, 10}, then are the following statements true or false?
(a) 5  M
(b) 11 N
(c) 2  M
(d) 8  N
Sets like M and N above have finite numbers of elements, and thus are called
.
We can count the number of elements in a finite set, and would use the notation n  M  to represent the
number of elements in set M .
It is also possible to have an empty set (also known as a null set) that contains no elements, and this kind of
set is notated as {} or  .
Sets like
,
,
, and
have infinitely many members, and thus are called
e.g. If N  {3, 6, 9, 10}, then n  N  
e.g. n   
.
e.g. n 
.

.
We can use set builder notation to represent some sets of numbers. A  x | x  , 2  x  4 means
“the set of all x such that x is an integer between 2 and 4 , including 2 and 4 ”.
e.g. Given A  x | x  , 2  x  7 ,
(a) list the elements of set A
(b) find n  A
(c) is it true or false that 2, 4,7  A ?
e.g. Given B  x | x  , x  3 AND x  1 ,
(a) list the elements of set B
(c) what is another way of notating set B ?
(b) find n  B 
.
Lesson 1.1 (continued)
P  Q is the notation for the
of sets P and Q , and is the set made up of all elements
which are in P or Q (or both).
P  Q is the notation for the
of sets P and Q , and is the set made up of all elements
which are in both P and Q .
If two sets have no common elements, then their intersection is the empty set and we say that the sets are
(or
).
e.g. Given M  2,3,5,7,8,9 and N  3, 4,6,9,10 ,
(a) list the elements of M  N
(b) list the elements of M  N
(c) Are M and N disjoint sets? Explain.
The symbol U is used to represent a universal set. If set A is a subset of this universal set (so A  U ),
of set A , denoted as A , is the set made up of all elements of U that are not in
then the
set A .
e.g. A  A 
e.g. n  A  
e.g. A  A 
e.g. If U  x | x  , 5  x  5 , A  x | x  , 3  x  3 , B  4, 3,0, 2 , and C  1 ,
list the elements of:
(a) A
(b) B 
(c) A  B
(d) A  B
(e) A  B
(f) A  B
(g)
 A  B 
(h)
 A  B 
(i)
 A 
(j)
 A  B   C