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
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$\;3,6,9,12,\ldots\;$
a set?
Solution:
Yes.
Is the number (say)
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$\,35{,}983{,}205{,}119{,}780{,}238{,}482{,}108{,}222\,$ 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
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$\,n\,$ members, where
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$\,n\,$ 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:
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$\{4,10\}\,$ 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
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$\,\{1,2,3,\ldots\}\,$ is an infinite set.
The number $\,7\,$ is an element of the set.
The number
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$\,\frac{30}{2}\,$ 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:

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

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

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$[\;\;]\;$ 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
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$\;\{\;\;\}\;$ are used for list notation.
 Separate members of the set with commas.
 Use three dots
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$\;\ldots\;$ 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:
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$\{1,2,3\}$
or
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$\{1,3,2\}$
or
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$\{2,1,3\}$
or
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$\{2,3,1\}$
or
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$\{3,1,2\}$
or
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$\{3,2,1\}$
The infinite set
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$\;\{0,1,2,3,\ldots\}\;$
contains all the whole numbers.
The finite set
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$\;\{0,1,2,\ldots,1000\}\;$
contains all the whole numbers between $\,0\,$ and $\,1000\,$.
The infinite set
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$\{2,1,0,\ldots\}\;$
contains all the integers that are less than or equal to $\;2\;$.
These are very different sets!
MEMBERSHIP IN A SET:

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

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

The sentence
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$\;x\notin S\;$
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
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$\,S\,$ is the set
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$\,\{1,2,3\}\,$,
then all of the following sentences are true:
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$\,1\in S\,$ and
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$\,3\in S\,$ and
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$\,4\notin S\,$ and
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$\,\frac{8}{4}\in S\,$ and
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$\,74\in S\,$
A SPECIAL SENTENCE:
The sentence
‘ Let
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$\,S = \{1,2,3\}\,$’
is used to assign the name
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$\,S\,$ to the set
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$\,\{1,2,3\}\,$.
The word ‘let’ is the key!
EXAMPLE:
Question:
How would a mathematician say:
“Take the set
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$\,\{a,b,c,d\}\,$
and give it the name
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$\,T\;$”?
Solution:
Let
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$\,T = \{a,b,c,d\}\;$.
(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:
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$\;\emptyset\;$ or
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$\;\{\;\}$
So, the empty set is empty! It has no members!
Be careful: the set
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$\,\{\emptyset\}\;$
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
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$\;(\;\;)\;$ are used when an endpoint is not included.
 Brackets
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$\;[\;\;]\;$
are used when an endpoint is included.
 The ‘infinity’ symbol
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$\,\infty\,$ is used to denote that an interval
extends forever to the right.
 The ‘negative infinity’ symbol
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$\,\infty\,$ 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
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$\,S = \{a,b\}\,$.
Then:

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

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

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

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

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

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$\,B\,$ is the empty set
 each member of
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$\,B\,$ 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