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Grade 11 Math Studies - 1. Sets and Venn Diagrams

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Steve Myers

on 7 October 2013

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Transcript of Grade 11 Math Studies - 1. Sets and Venn Diagrams

Grade 11 Math Studies
Sets and Venn Diagrams
Sets and Venn Diagrams
Opening Problem:
A baseball team of 21 players has 15 players that bat right handed and 6 that are ambidextrous. There are 3 pitchers who do not bat.
How many players bat only left handed?
How many players bat only right handed?
Word of the Day
Set Notation
Set - a collection of objects or things
We represent a set with a capital letter OR with curly brackets {}
For example:
A = the set of all students in grade 11 at Riverside
E = {0,2,4,6,8}
R = {red, orange, yellow, green, blue, indigo, violet}
Elements
Elements are the objects or things that make up a set.
- "is an element of"
- "is NOT an element of"
So, if
, then
and
.
Subsets
Quick Check:
P is the set of all odd numbers less than or equal to 13.
List the elements of P.
True or false:
A is a subset of B if all elements in A are also in B.
A is a proper subset of B if it is a subset of B, but NOT EQUAL to B.
- "A is a subset of B."
- "A is a proper subset of B."
What's the difference between subset and proper subset?
The definition of subset says "If A is a subset of B, then all elements of A are also in B."
By this definition, could A be exactly the same set as B?
The definition of PROPER subset excludes the possibility of the two sets being equal. In other words, if .
then
If A = {1,2,3}, list all possible subsets of A. Which are proper subsets?
For example:
If B = {-2,-1,0,1,2}, and A = {-2,0,2},
then .
** An element can only appear in a set ONCE. Don't write the same number or object twice.
Ex: Don't write {2,0,1,2,2}. Simply write {0,1,2}.
TWO IMPORTANT SETS:
The universal set - the set of all elements under consideration.
The empty set - the set with no elements.
- the universal set
- the empty set
The Complement of a Set
The complement of a set A is the set of all the elements in the universal set that are NOT in A.
- the complement of A.
No element in A can be in and no element in can be in A. So if B is the set of all boys in this classroom, then what would be?
Take a few minutes to practice! Work on Exercise 3A on p. 98.
PRACTICE
Special Number Sets
Natural numbers -
Integers -
Positive Integers -
Rational Numbers -
Irrational Numbers
Real Numbers - the set of all
numbers on the number line
Set Builder Notation
"The set of all m over n such that m and n belong to the set of integers."
Take a few minutes to practice! Work on Exercise 3B.1 and 3B.2 on p. 99
PRACTICE
Set Builder Notation
We can easily describe sets of numbers using only symbols using interval notation:
"The set of all x such that x lies between negative three and two, including two, and x is a real number."
In words:
On the number line:
If we want to talk about a different set of numbers, we can:
Set Builder Notation (cont'd)
In words:
On the number line:
"The set of all integers x such that x lies between negative five and five."
Example 3
Example 4
PRACTICE
Take a few minutes to practice! Work on Exercise 3C on pg. 102 in your textbook.
Venn Diagrams
A Venn diagram is a visual representation of sets. We use a rectangle for the universal set and circles for any other set.
Example 5
Subsets: Circles Within Circles
If B is a subset of A, we make a circle for A completely inside B:
Intersecting Sets: Overlapping Circles
If two sets have elements in common, but they are not subsets of each other, then we use overlapping circles:
PRACTICE
Take a few minutes to practice! Work on Exercise 3D on pg. 103 in your textbook.
Union and Intersection
The union of A and B is the set of all elements in A OR B OR both.
The intersection of A and B is the set of all elements in both A AND B.
- the union of A and B.
- the intersection of A and B.
Quick Check:
List the elements of :
a) b)
Example 8
Set Identities
Big Idea:
These identities are always true:
We can prove these using Venn diagrams!
PRACTICE
Take a few minutes to practice. Work on Exercise 3E starting on pg. 107 in your textbook.
Problem Solving
With Venn Diagrams

When problem solving, we use a number in brackets to represent the number of elements in a region.
- the number of elements in the set A.
and
Finite and Infinite Sets
Finite set -
A set with a finite number of elements
Infinite Set -
A set with infinitely many elements
is a finite set.
The set of all even numbers is an infinite set.
Irrational Numbers
Irrational numbers CANNOT be written in the form . Here's an easy way to identify irrational numbers:
Non-terminating, non-repeating decimals
Examples:
Example 1
Example 2
Relationships with Complementary Sets
Three obvious relationships that arise are:
Big Idea: Subsets and Proper Subsets
In Set Builder Notation
In Set Builder Notation:
Example 6
Example 7
Union of Sets
- A OR B OR BOTH
Disjoint Sets
Disjoint sets have no elements in common.
Disjoint Sets
Example 9
PRACTICE
Take a few minutes to practice! Work on Exercise 3C on pg. 102 in your textbook.
Example 10
Example 11
Example 12
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