Introduction to Sets

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, FIGH-night).

If a set is not finite, then it is infinite (pronounced with a short i, IN-fi-nit).

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:

$\{\;\;\}\;$ are called braces.
They are used in list notation for sets (see below).
$(\;\;)\;$ are called parentheses. (Singular form is parenthesis.)
They are used in interval notation for sets (see below).
$[\;\;]\;$ 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:

$\,S\,$ 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

$\,7-4\in S\,$

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

(MAX is 24; there are 24 different problem types.)