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 [beautiful math coming... please be patient] $\;3,6,9,12,\ldots\;$ a set?
Solution: Yes.
Is the number (say) [beautiful math coming... please be patient] $\,35{,}983{,}205{,}119{,}780{,}238{,}482{,}108{,}222\,$ 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:

DEFINITIONS elements, members;   finite set;   infinite set
The objects in a set are called its elements or its members.

If a set has [beautiful math coming... please be patient] $\,n\,$ members, where [beautiful math coming... please be patient] $\,n\,$ 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:
[beautiful math coming... please be patient] $\{4,10\}\,$ 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 [beautiful math coming... please be patient] $\,\{1,2,3,\ldots\}\,$ is an infinite set.
The number $\,7\,$ is an element of the set.
The number [beautiful math coming... please be patient] $\,\frac{30}{2}\,$ 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:

LIST NOTATION FOR SETS:

EXAMPLES:
Here are six names for the same finite set:
[beautiful math coming... please be patient] $\{1,2,3\}$    or    [beautiful math coming... please be patient] $\{1,3,2\}$    or    [beautiful math coming... please be patient] $\{2,1,3\}$    or    [beautiful math coming... please be patient] $\{2,3,1\}$    or    [beautiful math coming... please be patient] $\{3,1,2\}$    or    [beautiful math coming... please be patient] $\{3,2,1\}$
The infinite set [beautiful math coming... please be patient] $\;\{0,1,2,3,\ldots\}\;$ contains all the whole numbers.
The finite set [beautiful math coming... please be patient] $\;\{0,1,2,\ldots,1000\}\;$ contains all the whole numbers between $\,0\,$ and $\,1000\,$.
The infinite set [beautiful math coming... please be patient] $\{2,1,0,\ldots\}\;$ contains all the integers that are less than or equal to $\;2\;$.
These are very different sets!
MEMBERSHIP IN A SET:

EXAMPLE:
If [beautiful math coming... please be patient] $\,S\,$ is the set [beautiful math coming... please be patient] $\,\{1,2,3\}\,$, then all of the following sentences are true:

[beautiful math coming... please be patient] $\,1\in S\,$   and   [beautiful math coming... please be patient] $\,3\in S\,$   and   [beautiful math coming... please be patient] $\,4\notin S\,$   and   [beautiful math coming... please be patient] $\,\frac{8}{4}\in S\,$   and   [beautiful math coming... please be patient] $\,7-4\in S\,$
A SPECIAL SENTENCE:

The sentence ‘ Let [beautiful math coming... please be patient] $\,S = \{1,2,3\}\,$’ is used to  assign  the name [beautiful math coming... please be patient] $\,S\,$ to the set [beautiful math coming... please be patient] $\,\{1,2,3\}\,$.
The word  ‘let’  is the key!

EXAMPLE:
Question: How would a mathematician say:
“Take the set [beautiful math coming... please be patient] $\,\{a,b,c,d\}\,$ and give it the name [beautiful math coming... please be patient] $\,T\;$”?
Solution: Let [beautiful math coming... please be patient] $\,T = \{a,b,c,d\}\;$.
(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: [beautiful math coming... please be patient] $\;\emptyset\;$ or [beautiful math coming... please be patient] $\;\{\;\}$

So, the empty set is empty! It has no members!

Be careful: the set [beautiful math coming... please be patient] $\,\{\emptyset\}\;$ is a set with one member—it is NOT the empty set.

INTERVAL NOTATION:

interval notation
SUBSETS:

Roughly, a subcollection from a set is called a subset.

EXAMPLE:
Let [beautiful math coming... please be patient] $\,S = \{a,b\}\,$.
Then:
DEFINITION subset
Let [beautiful math coming... please be patient] $\,S\,$ be a set.
Set [beautiful math coming... please be patient] $\,B\,$ is called a subset of $\,S\,$ if any one of the following three conditions holds:
  • [beautiful math coming... please be patient] $\,B\,$ is the set $\,S\,$ itself
  • [beautiful math coming... please be patient] $\,B\,$ is the empty set
  • each member of [beautiful math coming... please be patient] $\,B\,$ 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
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
AVAILABLE MASTERED IN PROGRESS

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