**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

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