Chapter 2
Speaking
Mathematically
Lesson 5:
The Language of Sets
Notation

Example
Using the Set-Roster Notation
a.
Let 𝐴 = 1,2,3 , 𝐵 = 3,2,1 , 𝑎𝑛𝑑 𝐶 = {1,1,2,3,3,3} What are the
elements of A, B, and C? How are A, B, and C related?
b.
Is 0 = 0?
c.
How many elements are in the set 1, {1} ?
d.
For each nonnegative integer 𝑛, let 𝑈𝑛 = 𝑛, −𝑛 . Find 𝑈1 , 𝑈2 ,
and 𝑈0 .
Set-Builder Notation
Let 𝑆 denote a set and Certain sets of numbers
are so frequently referred to that they are given
special symbolic names.
Symbol
Set
Set of all real numbers
set of all integers
set of all rational numbers, or quotients of integers
set of nonnegative integers/ set of natural numbers

Addition of a superscript + or - or the letters nonneg
indicates that only the positive or negative or
nonnegative elements of the set, respectively, are to
be included.

The set of real numbers is usually pictured as the set of all
points on a line. Each point to the right of the origin
corresponds to a positive real nmber. Each point to the
left of the origin corresponds to a negative real number
and put a minus sign in front of the number.

The set of integers corresponds to a collection of points
located at fixed intervals along the real number line.
Set-builder Notation
Let S denote a set and let P(x) be a property that
elements of S may or may not satisfy. We may define a
new set to be the set of all elements x in S such that
P(x) is true. We denote this as follows:
{x  S P ( x )}
the set of all
such that
Example:
Given that R denotes the set of all real
numbers, Z the set of all integers, and Z+ the
set of all positive integers, describe each of the
following sets.
a. {x ϵ R I -2 < x < 5}
b. {x ϵ Z I -2 < x < 5}
c. {x ϵ Z+ I -2 < x < 5}
Subsets

A basic relation between sets is subset.

If A and B are sets, then A is called a subset of B, written
of B.
if, and only if, every element of A is also an element
Symbolically:
means that, for all elements x, if
and

The phrases A is contained in B and B contains A are
alternative ways of saying that A is a subset of B.

means that There is at least one element x such that
and
.
Proper Subset
U
Let A and B be sets. A is a proper subset of B if, and
only if, every element of A is in B but there is at least one
element of B that is not in A. Denoted by A B.
Ordered Pair
Given elements a and b, the symbol (a,b) denoted the
ordered pair consisting of a and b together with the
specification that a is the first element of the pair and b is
the second element. Two ordered pairs (a,b) and (c,d)
are equal if, and only if, a=c and b=d. Symbolically:
(a,b) = (c,d) means that a=c and b=d.
Example:
Let A = Z+ , B = {n ϵ Z I 10 < n < 100} and C = {100, 200, 300, 400, 500}
Determine the truth value.
2. C
U U
1. B
A
A
3. C and B have at least one element in common
5. C
U U
4. C
B
C
Cartesian Product
Given sets A and B, the Cartesian product of A
and B, denoted by
and read as “A cross B”, is
the set of all ordered pairs (a,b) where a is in A and
b is in B.
Symbolically:
Example:
Let A = {1, 2, 3} and B = {u, v}.
1. Find A x B ={(1,u),(1,v),(2,u),(2,v),(3,u),(3,v)}
2. Find B x A.
3. Find B x B.
4. How many elements are in A x B, B x A, and B x B?
5. Describe R x R, where R is the set of all real
numbers.
Exercises 2.1
A.Let P = {x ϵ Z I 3 < x < 9}, Y = {x ϵ Z - I x is even and
greater than -8}
1. Describe the statement using set roster notation
2. Find the cartesian product of the sets
B. Given that R denotes the set of all real numbers, Z the set of
all integers, and Z+ the set of all positive integers, describe each
of the following sets.
a. {x ϵ R I -5 < x < 1}
b. {x ϵ Z I -5 < x < 1}
c. {x ϵ Z- I -5 < x < 1}
Exercises 2.2
C. Let A = {1, 2, 3} and B = {u, v}.
1. Find A x B.
2. Find B x A.
3. Find B x B.
4. How many elements are in A x B, B x A, and B x B?
5. Describe R x R, where R is the set of all real numbers.
Exercises 2.2
Let A = {2, {2},(√2)2} , B = {2, {2}, {{2}}} and
C = {2}
Determine the truth value.
U
B
2. B
U
A
3. A
U
B
4. C
U
B
U
A
1. A
5. C