Sets
We won’t spend much time on the material from this and the next two
chapters, Functions and Inverse Functions. That’s because these three chapters are mostly a review of some of the math that’s a prerequisite for this
course. You don’t need to know all of it, but you should know much of
it. Identify the topics in these three chapters that you are uncomfortable
with, and spend the time to catch up on those topics. Ask your classmates,
instructor, or perhaps a tutor to help you out.
Sets
A set is a collection of objects. For example, the set of countries in North
America is a set that contains 3 objects: Canada, Mexico, and the United
States.
Set notation. Writing {2, 3, 5} is a shorthand for the set that contains the
numbers 2, 3, and 5, and no objects other than 2, 3, and 5.
The order in which the objects of a set are written doesn’t matter. For
example, {5, 2, 3} and {2, 3, 5} are the same set. Alternatively, the previous
sentence could be written as “For example, {5, 2, 3} = {2, 3, 5}.”
If B is a set, and x is an object contained in B, we write x 2 B. If both x
and y are objects contained in B then we write x, y 2 B. If x is not contained
in B then we write x 2
/ B.
Examples.
• 5 2 {2, 3, 5}
• 2, 3 2 {2, 3, 5}
•12
/ {2, 3, 5}
Subsets. One set is a subset of another set if every object in the first set is
an object of the second set as well. The set of weekdays is a subset of the set
of days of the week, since every weekday is a day of the week.
A more succinct way to express the concept of a subset is as follows:
The set B is a subset of the set C if every b 2 B
is also contained in C.
11
Writing B ✓ C is a shorthand for writing “B is a subset of C ”. Writing
B * C is a shorthand for writing “B is not a subset of C ”.
Examples.
• {2, 3} ✓ {2, 3, 5}
• {2, 3, 5} * {3, 5, 7}
Set minus. If A and B are sets, we can create a new set named A B
(spoken as “A minus B”) by starting with the set A and removing all of the
objects from A that are also contained in the set B.
Examples.
• {1, 7, 8}
{7} = {1, 8}
• {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
*
*
*
*
*
{2, 4, 6, 8, 10} = {1, 3, 5, 7, 9}
*
*
*
*
*
*
*
*
Numbers
Among the most common sets appearing in math are sets of numbers.
There are many di↵erent kinds of numbers. Below is a list of some that are
important for this course.
Natural numbers. N = {1, 2, 3, 4, ...}
Integers. Z = {..., 2, 1, 0, 1, 2, 3, ...}
Rational numbers. Q is the set of fractions of integers. That is, the
n
numbers contained in Q are exactly those of the form m
where n and m are
integers and m 6= 0.
For example, 13 2 Q and 127 2 Q.
Real numbers. R is the set of numbers that can be used to measure a
distance or magnitude, or the negative of a number used to measure a distance
12
Rational
thesetsetofoffractions
fractions
integers.
is, the
Rational numbers.
numbers. QQ isis the
of of
integers.
ThatThat
is, the
n
numbers
in QQ are
areexactly
exactlythose
thoseof of
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m are
numbers contained
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thethe
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n and
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m where
integers
integers and
and m
in ⌅=
01 0.0.
For
Q and
and ~127E⇤Q~Q.
For example,
example, ~3 ⇤
E Q
or magnitude.
magnitude. The
The set
set ofof real
real numbers
numbers can
can be
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as aa line
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called “the
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i.W
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lIT
⇧~ and ir are two of very many real numbers that are not rational numbers.
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Numbers
as
subsets.
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an integer.
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Any integer
integer isis aa
Numbers
as
subsets.
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notion of what a real number natural
is.)
rational
number,
and any
any
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number
real number.
number. We
We can
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write
rational
number,
and
isis aa real
Numbers
as subsets.
N çrational
Z ç Q çnumber
JR
thismore
moresuccinctly
succinctly
as N ⇥ Z ⇥ Q ⇥ R
Numbers
as subsets.
this
as
N*✓✓ZZ* ✓✓QQ
✓R
*
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* N
* ✓
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More set
set notation
notation
More
Sometimes in
in math
math we
we find
find itit more
more precise
precise to
to use
use aa common
common collection
collection ofof
Sometimes
symbols to
to express
express our
our thoughts.
thoughts. The
The symbols
symbols have
have less
less room
room for
for interpreinterpresymbols
tation than
than normal
normal written
written language,
language, so
so less
less confusion
confusion arises
arises when
when we
we use
use
tation
them. One
One important
important example
example ofof this
this isis aa method
method we
we use
use to
to describe
describe sets.
sets.
them.
Let’sstart
startwith
withan
anexample,
example,then
thenwe’ll
we’llexplain
explainthe
theexample:
example:
Let’s
nn nn
oo
n,m
m22ZZand
andm
m6=6=00
QQ==
n,
m
m
Let’s break
break down
down this
this equation
equation one
one symbol
symbol at
at aa time.
time. We
We know
know that
that QQ isis
Let’s
theset
setofofrational
rationalnumbers,
numbers,and
andthe
theequal
equalsign
signmeans
meansthat
thatQQisisthe
thesame
sameset
set
the
asthe
theset
seton
onthe
theright
rightofofthe
theequal
equalsign.
sign.
as
Youknow
knowthat
thateverything
everythingon
onthe
theright
rightofofthe
theequal
equalsign
signisisaaset
setbecause
becauseit’s
it’s
You
all surrounded
surrounded by
by the
the two
two curly
curly brackets
brackets “{”
“{” and
and “}”,
“}”, so
so now
now let’s
let’s discuss
discuss
all
everything in
in between
between the
the curly
curly brackets,
brackets, moving
moving from
from left
left to
to right.
right. The
The
everything
n
n means that the set on the right side of the equal sign is a set of
fraction m
fraction
m means that the set on the right side of the equal sign is a set of
nn means “that satisfy the
fractions. The
The vertical
vertical line
line immediately
immediately following
following m
fractions.
m means “that satisfy the
12
13
following rules”. Everything to the right of the vertical line is the collection
of rules that the fractions of the set satisfy, in this case, n and m are integers,
n
and m 6= 0. So the set m
| n, m 2 Z and m 6= 0 is the set of all fractions
n
m that satisfy the rules that n and m are integers and m 6= 0.
Putting it all together, the expression
nn
o
Q=
n, m 2 Z and m 6= 0
m
should be read as saying that the set of rational numbers is the set of fractions
of integers, as long as the denominator of the fractions do not equal 0. And
we already knew that. This is perhaps just a slightly more efficient way to
write that. That is, it will be more efficient once you are comfortable with
it, and that won’t take too long.
Examples.
n
•Z= m
| n 2 Z and m = 1 says that the integers are the same set
as the set of fractions that satisfy the rules that the numerator is an integer
and the denominator equals 1, and that’s true. Notice that 31 = 3 and that
3 2 Z. Similarly, 117 = 17 and 17 2 Z, etc.
• E = n 2 Z | n2 2 Z says that E is the set of integers that satisfy
the rule that when you divide them by 2, you still have an integer. Notice
that 42 = 2 2 Z, that 218 = 9 2 Z, and that 32 2
/ Z. So the above equation
of sets is telling us is that E is the set of even numbers.
• N = n 2 Z | n > 0 says that the natural numbers is the set
of all integers that are positive. And that’s true. The positive numbers in
{..., 2, 1, 0, 1, 2, 3, ...} are exactly the numbers {1, 2, 3, 4, ...}.
• { x 2 {2, 7, 3} | x < 0 } = { 3} says that
number from the collection of 2, 7, and 3.
*
*
*
*
*
*
*
*
*
*
3 is the only negative
*
*
*
Intervals
Below is a list of 8 important types of subsets of R. In the sets below, a
and b are real numbers. That is, a, b 2 R. Also, in the sets below a  b. Any
set of one of these types is called an interval.
14
a xx 
x bb }}b
[a,{a,b]
b] =
= {=
{ xx{x
2R
R |RI
| aa 


[a,
b]
2
}
(
b
Q
a x<x
(a,(a,b)
b) =
==
2R
R |RI
| aa <
<
x<
< b<
b }}b} .(
(a,
b)
{{ xx{x
2
Oj.rn1u,imuwsvemO
b
0..
[a,[a,b)={xERIa<x<b}
b) =
= {{ xx 2
2R
R || aa 
 xx <
< bb }}
(
[a,
b)
411.IJIL.*1L_1*_O
0..
a x<x
(a,(a,b]
b] =
==
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| aa <
<
x
 bb }}b}
(a,
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{{ xx{x
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a xx }}x}
[a,[a,oo)={x
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= {{ xx 2
2R
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
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ERI
b
b
M$W4*W
1, b]
b] =
= {{ xx 2
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(( (-oo,b]={xERIxb}
1,
b
(a,(a,oo)={XERIa<x}
1) =
= {{ xx 2
2R
R || aa <
< xx }}
(a,
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(_
0
1, b)
b) =
= {{ xx 2
2R
R || xx <
< bb }}
(( (—oo,b)={XERIX<b}
1,
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TheEmpty
The
Empty
Set
Set
EmptySet
The
The
setset
{4,{4,
7, 9}
9}9}
is is
the
setset
that
contains
thethe
numbers
4, 4,
7, 7,
and
9. 9.The
The
setset
The
the
that
contains
7,
The
set
{4,
7,
is
the
set
that
contains
the
numbers
4,
7,
and
9.
set
numbers
and
The
{4,{4,
7}7}contains
contains
only
7, 7,the
the
setset{4}
{4}
contains
only
4, 4,and
and
thethe
setset{{ }}{ }
only44 and
4and
and7,
containsonly
{4,
7}
set
only
4,
the
set
the
{4}contains
contains
only
and
contains,
well,
nothing.
contains,
nothing.
well,
contains,
well,
nothing.
It It
probably
seems
silly
toto
you
now,
but
the
setset
that
has
nothing
in in
it,it,
that
probably
seems
silly
now,
It
probably
seems
silly
to
you
now,
but
the
set
that
has
nothing
in
it,
that
you
but
the
that
has
nothing
that
is is
the
setset
},
anan
important
example
of of
set,
and
it’sit’s
set
that
comes
upup
the
important
is
the
set
{{ },
important
example
of
aa set,
and
it’s
aa set
that
comes
up
example
a set,
and
a set
that
comes
{ },isis isan
frequently
in inmath.
math.
It Itcomes
comes
upup
often
enough
that
we’ve
given
it it
name.
math.It
frequently
comes
frequently
in
up
often
enough
that
we’ve
given
it
aa name.
often
enough
that
we’ve
given
a name.
We
call
it it
thethe
empty
set,
because
it’sit’s
thethe
setset
that
has
nothing
in in
it.it.
call
We
empty
set,
because
We
call
it
the
empty
set,
because
it’s
the
set
that
has
nothing
in
it.
nothing
that
has
Even
more
than
having
own
name,
we’ve
given
itsits
own
symbol.
That’s
having
Even
more
In
addition
tothan
having
itsit’s
own
name,
we’ve
given
ititits
own
symbol.
it’s
own
name,
we’ve
given
it
own
symbol.That’s
That’s
because
wewewrote
wrote
the
empty
setsetas
asas{{ },
},
then
readers
might
just
wonder
becauseifif if
wrotethe
theempty
emptyset
because
we
readers
then
readersmight
mightjust
justwonder
wonder
{ },then
whether
that’s
typo,
that
perhaps
wewe
intended
toto
type
something
inside
of of
whether
whether
that’s
aa typo,
that
perhaps
we
intended
to
type
something
inside
of
that’s
that
a typo,
perhaps
intended
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something
inside
the
curly
brackets,
but
just
neglected
to.to.Instead,
Instead,
wewe
useuse
the
symbol
to
the
brackets,but
butjust
curly
the
curly
brackets,
neglected
to.
use
the
symbol
;; to
just
neglected
Instead,we
the
symbol
0 to
refer
to
the
empty
set.
refer
empty
set.
refer to to
thethe
empty set.
14 14
15
Example.
If S was the set of all roots of the quadratic x2 4, and then S = { 2, 2}.
If S was the set of all roots of the quadratic x2 , then S = {0}.
And if S was the set of all real number roots of the quadratic polynomial
2
x + 1, then S = ;, because x2 + 1 has no real number roots.
The Empty Set is a subset of any other set
In the explanation a few paragraphs before, we began with the set {4, 7, 9}.
We took away an object and were left with the smaller set {4, 7}. Then we
took away another object and were left with the still smaller set {4}. We
ended by taking away the last object to arrive at a still smaller set, namely
the empty set, ;.
We can think of a set B as being a subset of another set C if we can remove
objects from C to obtain B. In that way we can write
; ✓ {4} ✓ {4, 7} ✓ {4, 7, 9}
Of course if C is any set, it doesn’t matter which one, and if we took away
all of the objects from C, then we’d be left with ;. That is to say,
regardless of what the set C is.
;✓C
16
Exercises
Determine whether the statements made in #1-15 are true or false.
1.) 3 2 {3, 6, 1}
2.) 1, 6 2 {3, 6, 1}
3.) 4 2 {3, 6, 1}
4.) 4 2
/ {3, 6, 1}
5.) {2, 3} ✓ {3, 6, 1}
6.) {2, 3} * {3, 6, 1}
7.) ; ✓ {3, 6, 1}
8.) {3, 6, 1}
9.) [1, 3) ✓ [1, 3]
10.) 2 2 [1, 3]
11.) 1 2 (1, 3]
12.) 3 2 (1, 3]
13.) [1, 3) ✓ (1, 1)
14.) [1, 3) ✓ ( 1, 5]
{6} = {3, 1}
15.) ; ✓ [1, 3)
Each of the sets in #16-21 is an interval. Name the interval in each of these
problems.
16.) [2, 5)
(3, 5)
17.) [2, 5)
(2, 5)
18.) [2, 5)
[4, 5)
19.) [2, 5)
{2}
20.) [2, 5)
[2, 3)
21.) [2, 5)
[2, 4]
17
Let T = {1, 2, 3}. Match the numbered sets on the left to the lettered sets
on the right that they equal.
22.) { x 2 T |
x
2
2 Z}
A.) {1, 2, 3}
23.) { x 2 T | x 2 N }
B.) {1, 2}
24.) { x 2 T | x > 1 }
C.) {1, 3}
2 Z}
D.) {2, 3}
x 2 N}
E.) {1}
2
/ Z}
F.) {2}
28.) { x 2 T | x  2 }
G.) {3}
25.) { x 2 T |
x
3
26.) { x 2 T |
27.) { x 2 T |
29.) { x 2 T |
x
2
1
x
2 Z}
H.) ;
18