Download 01-12 Intro, 2.1 Sets

Welcome to
Mathematics for
Elementary Educators!
1/12/16
Section 2.1: Sets
Today We’ll Discuss
What is a mathematical set?
How can the idea of cardinality be used to
describe numbers?
How can sets be used to develop
inequalities?
What is a Number?
How would you explain to a young student
what a “number” is???
Defining a “Set”
The word “set” is used commonly in the
English language.
How would you define a “set” to a young
student?
Defining a Set
Definitions: A set is a collection of objects.
Each object in a set is called an element
of that set.
We usually place all the elements of a set
in “squiggly” brackets:
{ }
There are two ways of describing sets:
listing method and set-builder notation
Notation - Listing Method
In the listing method, we list out every
element in the set.
For example,
A = {1, 2, 3, 4, 5}
B = {7, 9, 11}
Notation - Set Builder
In the set builder notation, we give a
placeholder variable and a defining quality
of the elements of the set.
For example,
A = {1, 2, 3, 4, 5}
A = {x|x is a whole number from 1 to 5}
Notation - Set Builder
In the set builder notation, we give a
placeholder variable and a defining quality
of the elements of the set.
For example,
B = {7, 9, 11}
B = {x|x is an odd number between 6 and 12}
Example
Rewrite the following in set-builder
notation:
A = {11, 13, 15, 17, 19, 21, 23}
B = {0, 2, 4, 6, 8, 10, … }
C = {2}
D = {3, 6, 9, 12}
Properties of Sets
Definition: The null set (or empty set) is
the set with no elements, and is denoted
by { } or ∅.
Definition: Two sets are equal if and only if
they have the same elements. (Order of
elements does not matter.)
Example
Which of the following sets are equal?
A = {3, 5, 7}
B = {x|x is an odd number, 2 ≤ x, and x ≤ 10}
C = {x|x is a prime number between 2 and 10}
D = {x|x is a whole number between 2 and 10
not divisible by 2}
Cardinality
Definition: The cardinality of a set is the
number of unique elements of the set.
Sets can have finite or infinite cardinality.
If a set is finite, we usually say exactly how
many elements it contains.
Example
Tell whether the set is finite or infinite. If
the set is finite, then find its cardinality.
A = {a, b, c, d}
B = {2, 7, 12, 14, 16, 20}
C=∅
D = {x|x is an odd number}
Defining Natural and Whole
Numbers, the “Mathy” Way
Definition: The set of natural numbers is
the set of all cardinalities of nonempty
finite sets.
Definition: The set of whole numbers is the
set of all cardinalities of finite sets.
Defining Natural and Whole
Numbers, the “Mathy” Way
Definition: The set of natural numbers is
the set of all cardinalities of nonempty
finite sets.
Natural Numbers: {1, 2, 3, 4, … }
Definition: The set of whole numbers is the
set of all cardinalities of finite sets.
Whole Numbers: {0, 1, 2, 3, 4, … }
One-to-One Correspondence
An important concept in learning numbers
is what we call a one-to-one
correspondence.
When two sets have the same cardinality,
their elements can be placed in a one-toone correspondence. (Every element of one
set gets exactly one “buddy” from the
other.)
Subsets
Definition: A set A is a subset of a set B if
and only if every element of A is also an
element in B. We denote this as A ⊆ B.
Ex:
A = {a, b, c}
B = {a, b, c, d}
Subsets - Quick Notes
Two equal sets are subsets of each other
If a set is a subset which does not contain
at least one element of its parent set, it
is a proper subset. This is denoted A ⊂ B.
The null set is a subset of every set
Example
Tell whether the statement is true or false.
1. The set of natural numbers is a subset of the
set of whole numbers.
2. The set of natural numbers is a proper subset
of the set of whole numbers.
3. The set of odd numbers is a subset of the set
of natural numbers.
4. The set of whole numbers is a subset of the set
of natural numbers.
5. The set of squares of the whole numbers is a
subset of the set of even numbers.
Inequality (Whole Numbers Only)
One way to talk about inequalities with
whole numbers is by using cardinalities of
sets.
We simply count the number of elements
and see which one has more.
Inequality (Whole Numbers Only)
Let A have cardinality m and B have
cardinality n.
If A ⊆ B, then m ≤ n.
If A ⊂ B, then m < n.
Example
How could we use the concept of sets
below to explain how one number is
greater than another?
Homework:
Read pages 39-43
Do problems
#6,8,12,16,17,18,20,27,28,41
Due: 1/17/17 Beginning of Class!