4.1 Set operations

• Wiskunde
• Middelbare school
• vwo
• Leerjaar 4

Good afternoon!

Schedule: 

• Homework
• Set symbols whiteboard revision
• Sets and set notation 
• Practice examples 
• Start on the homework

Learning goals 

At the end of these two lessons I can:

• Categorize most numbers in a numerical set
• Represent sets and their operations 
  
• Find the intersection and union of sets 

Take out your diaries/notebook

Homework for Tuesday 10-1-2023:

                                         p. 223: All three questions of practice 1

Types of numbers

You have different types of numbers: 

Natural numbers (counting numbers): Positive whole numbers

{0, 1, 2, 3, 4, ...}

All natural numbers = 

ℕ

Types of numbers

You have different types of numbers: 

Integers: Positive and negative whole numbers

{..., -2, -1, 0, 1, 2, 3, ...}

                 , 

                                    All integers = 

Every natural number is an integer !

ℤ

Types of numbers

You have different types of numbers: 

Rational numbers: Numbers you can write as a fraction 

for example           ,              , 4

                 , 

                           All ration numbers= 

Every integer is a rational number, therefore every natural number is also a rational number !

ℚ

​4​​3​​

0.2

Types of numbers

You have different types of numbers: 

Real numbers : All rational and irrational numbers 

For example          ,           , -2, 0.9999999....

 

               All real numbers =

​4​​3​​

√​2​​​

ℝ

Types of numbers

1. Read through the 1st page of the handout. 

2. Each group will receive 2 notes with numbers on them. 

3. Discuss in your group to what numeric set(s) your numbers belong to. (some numbers belong to multiple sets, choose 1).

4. Stick the post-it note on the whiteboard in the set where you think it belongs. 

5. When you've finished, copy the chart on the whiteboard

Set notation

What is the Mathematical symbol for.... 

.. is an element of ..

Set notation

What is the Mathematical symbol for.... 

.. is an element of ..

∈

Set notation

What is the Mathematical symbol for.... 

.. is a subset of .. 

Set notation

What is the Mathematical symbol for.... 

.. is a subset of .. 

⊆

Set notation

What is the Mathematical notation/symbol for.... 

Listing a set 

Set notation

What is the Mathematical notation/symbol for.... 

Listing a set 

{  ..., ..., ...  } 

Set notation

What is the Mathematical notation/symbol for.... 

The empty set 

Set notation

What is the Mathematical notation/symbol for.... 

The empty set 

{ } or ∅

Review

Sets and set notation

A = { all even numbers between 1 and 10} 

A = { 2, 4, 6, 8, 10}

Set 

Description of the elements of A

Listing of the elements of A

Sets and set notation

if A = { -3, -2, 0, 1, 5}

Then

A+ = {.... }

A- = { ....}

A* = { .... }

Find the elements of these 3 sets.

Sets and set notation

if A = { -3, -2, 0, 1, 5}

Then

A+ = {1, 5}

A- = {-3, -2}

A* = {-3, -2, 1, 5}

Union and intersection

What are we having for dinner tonight? 

A is the set of meals that I like, B is the set of meals that my mom likes. 

A = {lasagna, burritos, quiche, pasta carbonara}

B = {chili, curry, quiche, lasagna}

Union and intersection

What are we having for dinner tonight? 

A is the set of meals that I like, B is the set of meals that my mom likes. 

A = {lasagna, burritos, quiche, pasta carbonara}

B = {chili, curry, quiche, lasagna}

A ∩ B = Intersection of A and B = The meals that both me and my mom likes.

Union and intersection

What are we having for dinner tonight? 

A is the set of meals that I like, B is the set of meals that my mom likes. 

A = {lasagna, burritos, quiche, pasta carbonara}

B = {chili, curry, quiche, lasagna}

A ∩ B = Intersection of A and B = The meals that both me and my mom likes.

A ∪ B = The union of A and B = The meals that either me or my mom likes. 

Universal and complementary set

The complementary set (notation: A' or Ac) of a set A contains all elements that DO NOT belong to A.

The universal set (notation: U) is a set which contains ALL ELEMENTS of a problem. 

Example: 

If U = { 1, 2, 3, 4, 5} and 

A = { 2, 4, 5} then 

A' = {1, 3} 

Set and set notation

Set and set notation

a. A∩B = The intersection of A and B

What elements do                    and                    have in common? None! 

 A∩B = ∅ (or { }) 

Set and set notation

b. A∪B' = The Union of A and the complement of B. 

B' = all elements in U that are not in B 

B' = {1, 4, 6, 8, 9, 10}

All elements of A and B' together: 

A∪B' = {1, 2, 4, 6, 8, 9, 10}        No double values in a set!

Set and set notation

c. (A' ∩ B)' = The complement of the intersection of A' and B. 

A' = {1, 3, 5, 7, 9, 10}

A' ∩ B = {3, 5, 7}

(A' ∩ B)'  = {1, 2, 4, 6, 8, 9, 10}

Set and set notation

d. (A ∪ B ∪ C)' = The complement of the union of A, B and C.  

A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

(A ∪ B ∪ C)'  = {10}

When looking for the compliment of a set, Compare the set with the Universal set and find the differences

Sets and set notation

 Read this as: 

U is the set of all x such that x is an integer larger or equal to 0 and smaller or equal than 20.

Listing of the elements: 

U = {0, 1, 2, 3, 4, 5, ... , 19, 20}

Homework: Practice 1

How would you describe U? List the elements of U. 

And how would you describe C? List the elements of C.

Homework: Practice 1

U = the set of all positive integers between (and including) 1 and 20

U = { 1, 2, 3, ..., 19, 20}

C = the set of all primes between (and including) 1 and 20

C = { 2, 3, 5, 7, 11, 13, 17, 19} 

Take out your diaries/notebook

Homework for Tuesday 10-1-2023:

                                         p. 223: All three questions of practice 1