DEFINITION
set
A set is a collection with the following property:
given any object, either the object
is in the collection, or isn't in the collection.
EXAMPLES:
Question:
Is ‘the collection of some people’ a set?
Solution:
‘The collection of some people’ is not a set; it is too vague.
Is your teacher in this
collection?
Maybe or maybe not!
Question:
Is the collection of numbers consisting of
$\;3,6,9,12,\ldots\;$
a set?
Solution:
Yes.
Is the number (say)
$\,35{,}983{,}205{,}119{,}780{,}238{,}482{,}108{,}222\,$ in this collection?
Well, either it is (if divisible by $\,3\,$)
or isn't (if not divisible by $\,3\,$).
Notice that it's not important whether you personally
know whether the answer is YES or NO;
all that matters is that the answer is definitely YES or NO.
NOTES ABOUT SETS:
 A set is a mathematical expression.
 Sets can have different names!
DEFINITIONS
elements, members; finite set; infinite set
The objects in a set are called its elements or its members.
If a set has
$\,n\,$ members, where
$\,n\,$ is a whole number, then it is called a finite set
(pronounced with a long i, FIGHnight).
If a set is not finite, then it is infinite
(pronounced with a short i, INfinit).
EXAMPLES:
$\{4,10\}\,$
is a finite set,
with two members.
The number $\,4\,$ is a member.
The number $\,10\,$ is a member.
(More on list notation for sets below.)
The set
$\,\{1,2,3,\ldots\}\,$
is an infinite set.
The number $\,7\,$ is an element of the set.
The number
$\,\frac{30}{2}\,$ is an element of the set.
(The name we use doesn't matter!)
The number $\,0.25\,$ is not an element of the set.
SYMBOLS USED IN CONNECTION WITH SETS:
The following symbols are used in connection with sets:

$\cssId{s50}{\{}\;\;\cssId{s51}{\}}\;$
are called
braces.
They are used in list notation for sets (see below).

$\cssId{s54}{(}\;\;\cssId{s55}{)}\;$
are called
parentheses.
(Singular form is parenthesis.)
They are used in interval notation for sets (see below).

$\cssId{s59}{[}\;\;\cssId{s60}{]}\;$
are called
brackets.
They are used in interval notation for sets (see below).
LIST NOTATION FOR SETS:
 List notation for sets is used whenever the elements of a set can be listed.

Braces
$\;\{\;\;\}\;$ are used for list notation.
 Separate members of the set with commas.

Use three dots
$\;\ldots\;$
to indicate that a pattern is to be repeated.
Be sure to list enough elements to clearly establish the pattern.
 For a finite set, the order that elements are listed doesn't matter.
EXAMPLES:
Here are six names for the same finite set:
$\{1,2,3\}$
or
$\{1,3,2\}$
or
$\{2,1,3\}$
or
$\{2,3,1\}$
or
$\{3,1,2\}$
or
$\{3,2,1\}$
The infinite set
$\;\{0,1,2,3,\ldots\}\;$
contains all the whole numbers.
The finite set
$\;\{0,1,2,\ldots,1000\}\;$
contains all the whole numbers between $\,0\,$ and $\,1000\,$.
The infinite set
$\{2,1,0,\ldots\}\;$
contains all the integers that are less than or equal to $\;2\;$.
These are very different sets!
MEMBERSHIP IN A SET:

The verb
$\;\in\;$
is used to denote membership
in a set.

The sentence
$\;x\in S\;$
is read as:
‘$\,x\,$ is in $\,S\;$’ or
‘$\,x\,$ is an element of $\,S\;$’ or
‘$\,x\,$ is a member of $\,S\;$’.

The sentence
$\;x\notin S\;$
is read as:
‘$\,x\,$ is not in $\,S\;$’ or
‘$\,x\,$ is not an element of $\,S\;$’ or
‘$\,x\,$ is not a member of $\,S\;$’.
EXAMPLE:
If
$\,S\,$ is the set
$\,\{1,2,3\}\,$,
then all of the following sentences are true:
$\,1\in S\,$ and
$\,3\in S\,$ and
$\,4\notin S\,$ and
$\,\frac{8}{4}\in S\,$ and
$\,74\in S\,$
A SPECIAL SENTENCE:
The sentence
‘ Let
$\,S = \{1,2,3\}\,$’
is used to assign the name
$\,S\,$ to the set
$\,\{1,2,3\}\,$.
The word ‘let’ is the key!
EXAMPLE:
Question:
How would a mathematician say:
“Take the set
$\,\{a,b,c,d\}\,$
and give it the name
$\,T\;$”?
Solution:
Let
$\,T = \{a,b,c,d\}\;$.
(The word ‘let’ is a vital part of this sentence!)
DEFINITION
empty set
The empty set is the unique set that has no members.
The empty set is denoted using either of these symbols:
$\;\emptyset\;$ or
$\;\{\;\}$
So, the empty set is empty!
It has no members!
Be careful:
the set
$\,\{\emptyset\}\;$
is a set with one member—it is NOT the empty set.
INTERVAL NOTATION:
 Interval notation is used to describe intervals of real numbers.
 Intervals of real numbers are infinite sets.

Parentheses
$\;(\;\;)\;$ are used when an endpoint is not included.

Brackets
$\;[\;\;]\;$
are used when an endpoint is included.

The ‘infinity’ symbol
$\,\infty\,$ is used to denote that an interval
extends forever to the right.

The ‘negative infinity’ symbol
$\,\infty\,$ is used to denote that an interval
extends forever to the left.
 The numbers used in interval notation always go from left to right on the number line.
SUBSETS:
Roughly, a subcollection from a set is called a subset.
EXAMPLE:
Let
$\,S = \{a,b\}\,$.
Then:

$\,\{a\}\,$
is a subset
(choose only the ‘$\,a\,$’ )

$\,\{b\}\,$
is a subset
(choose only the ‘$\,b\,$’ )

$\,\{a,b\}\,$
is a subset
(choose everything!)

$\,\{\;\}\,$
is a subset
(choose nothing!)
DEFINITION
subset
Let
$\,S\,$ be a set.
Set
$\,B\,$ is called a subset of $\,S\,$
if any one of the following three conditions holds:

$\,B\,$ is the set $\,S\,$ itself

$\,B\,$ is the empty set

each member of
$\,B\,$ is also a member
of $\,S\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Interval and List Notation