02
2
Todays Day Travel Plans
What did we talk about yesterday?
Sets
Equivalent Sets
Formally Defining Whole Numbers
Operation on Sets
Operations on the Whole Numbers
1
3
What did we talk about yesterday?
Mathematics is the _________ of Patterns
Find Patterns
Utilize to Solve __________
Analyze to Understand
Applied Math solves Real World Problems through
_____________
Defining ____________
4
What did we talk about yesterday?
Modeling takes a Complex Phenomenon and replaces
it with a _________Model
Usually we have some kind of _________ of the model
_________ depends on the questions one wishes to answer
Relationships
Name Common traits between two things
2
5
Sets
We will now Review some material about sets you may
or may not have seen before.
A Set is a _________ of a container
Real containers like cups, backpacks, cars, etc. have many
more distinct traits other than ability to put things in them.
Sets are abstract containers, the ability to contain things is
their only defining trait.
____________is written language used to describe
mathematical ideas
6
Sets Review Notation
The symbols { } and often times capital letters are used
to define [that is, name] sets
Sets are defined/described by
Roster Notation, i.e. Listing elements
The elipses mean a Recognized pattern continues on.
3
7
Sets Review Notation II
Sets are defined/described by
Description, with words
Set Builder notation, the use of mathematical symbols,
formulas, etc. to describe a set
Visually with a Euler or Venn diagram.
8
Set Containment
a set A is a _________of B if every member of A is a
member of B
A is a ________ ________ of B if B is not also a subset
of A
Consequence:
4
9
Sets and One-to-One Correspondence
A one-to-one correspondence between sets A and B is
a way of matching each thing in A with exactly one
thing from B and each thing in B with exactly one
thing in A.
10
Set Equivalence
Two sets are said to be equivalent if they can be put
into a one-to-one correspondence
Sets that are _________ are automatically equivalent.
Sets that are equivalent do not have to be _________.
5
11
Numbers
Numbers are Models for the idea that represent
How Much?
How Many?
More?
Less?
What Part?
Where?
How Close?
12
Whole Numbers
The numbers that occur naturally and the notion of
zero which historically is thought up some time after
the original counting numbers.
Informally, we take them for granted.
Today we will define them formally as sets
This is weird Using sets to model numbers so that we
can use numbers to model other things
6
13
Whole Numbers Defining
Does Nothing Exist?
Empty _________ ________
Empty __________
No _______
Let 0 = {}
Note that this is not the zero you know and love.
0 is being defined here as a name for the empty set.
Let 1 = {0}
Note, this is not the usual 1. It is a name for this set.
Does = 0 1?
14
Whole Numbers Defining II
So 0 = {}, 1 = {0}, 2 = {0,1}, 3 = {0, 1, 2}, 4 =
In general,
n = {0, 1, 2, , n-1}
This is known as a ____________ Definition, one for which
current cases are defined in terms of previous ones.
The process of defining the whole number symbols in this way is
called __________ Arithmetic
The expression n = is called the ____________ function.
7
15
Whole Numbers and Set Equivalence
Observe that all the following are equivalent, meaning that each
set can be put into a one-to-one correspondence with any other.
One of these sets is the one named _____, which one?
Each of these sets has ________things in it.
16
Cardinality of Sets
Informally: The number of members in a set.
Formally:
The ___________ of a set A, denoted |A|, is n if and only
if A is equivalent to the set named n.
Example:
Alternatively, we could have
8
17
Whole Numbers Defining III
Note that
is not a proper subset of any set n
The cardinality/Size of
is said to be
.
A set is said to be _______ if is equivalent to any set
n={0, 1, 2, , n-1} and _____________ otherwise.
This is actually the definition that makes the sets named
0, 1, 2, . synonomous with the whole numbers 0, 1, 2,
18
Operations
A _________operation is a way of combining two
things to create a new thing.
There are operations on sets like
Most people experience operations on numbers first like
An operation is _______ on a set if that operation of
any two members of the set is also a member of the
set
The only operations [of the basic four] that are closed on the
wholes numbers are
9
19
Two Basic Set Operations, Union
The union of two sets is the third set containing any
member of either parent set
Notation:
the cup symbol.
20
Two Basic Set Operations, Intersection
The intersection of two sets is the third set containing
only things which are in both of the parent sets.
Notation:
the cap symbol.
10
21
Arithmetic Operations, Addition
We just defined Whole numbers in terms of Sets.
Therefore, we might try to use set operations to define
arithmetic ones
Addition, +, on
If a, b are in
then a + b = c where
c is equivalent to
a is equivalent to
b is equivalent to
and
is equivalent to
22
Arithmetic Operations, Addition, Example
This is abstract but for a moment recall counting on your
fingers when you were in elementary school
Count to
on your left hand!
Count to
on your right hand!
Count all your fingers now!
11
23
Arithmetic Operations, Multiplication
Multiplication is defined initially in terms of _________
Multiplication, , on
If a, b are in
then a b =
Example:
24
What about the other operations?
It is possible to define subtraction and division on
whole number by using an _________ process.
12
25
Lack of Closure
Subtraction and division work for some pairs of whole
numbers
But not others, this is why it is not closed
13