Module SV: Sets and Venn Diagrams
Module SV Learning Objectives:
- Use correct set notation and terminology (SV.1)
- Determine if sets are equivalent and/or equal (SV.1)
- Determine if a set is well defined (SV.1)
- Determine if a set is finite or infinite (SV.1)
- Distinguish between a subset and a proper subset (SV.1)
- Determine the number of subsets and proper subsets of a set (SV.1)
- Identify the universal set (SV.2)
- Find intersection, unions, and complements of sets (SV.2)
- Model relationships between sets using Venn diagrams (SV.3)
- Recognize and be able to apply cardinality notation (SV.4)
- Use cardinality notation and formulas to solve Venn diagrams (SV.4)
Section SV.1 – Set Basics
Section SV.2 – Union, Intersection, and Complement
Section SV.3 – Venn Diagrams
Section SV.4 – Cardinality
Summary of Set Theory Symbols
|
Symbol |
Name |
Example |
Explanation |
|
{list of elements} |
Set |
A = {1, 3} B = {2, 3, 9} C = {3,9} |
Collection of distinct objects Curly brackets are used to designate a set |
|
∩ |
Intersection |
A ∩ B = {3} |
Belong to both set A AND set B |
|
∪ |
Union |
A ∪ B = {1,2,3,9} |
Belong to set A OR set B (or both) |
|
⊂ ⊃ |
Proper Subset |
1 ⊂ {1,2} C ⊂ B 1,2 ⊃ {1} |
A set that is contained in another set |
|
⊆ ⊇ |
Subset |
1 ⊆ A {1,3} ⊆ A |
A set that is contained in or equal to another set |
|
⊄
|
Not a proper subset |
{1,3} ⊄ A |
A set that is not contained in another set |
|
∈ |
Is a member or element |
3 ∈ A |
3 is an element in set A |
|
∉ |
Is not a member or element |
4 ∉ A |
4 is not an element in set A |
|
∅ or {} |
Empty Set |
{ } |
The set that contains no elements |
Section SV.1 – Set Basics
It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets to understand relationships between groups, and to analyze survey data.
An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.
| Set |
|
A set is a collection of distinct objects, called elements of the set. A set can be defined by describing the contents. This is called verbal notation. A set can also be defined by listing the elements of the set, enclosed in curly brackets. This is known as roster notation. |
| Example 1 |
|
Some examples of sets that are defined by describing the contents of the set: a) The set of all even numbers b) The set of all books written about travel to Chile |
|
Some examples of sets that are defined by listing the elements of the set: a) {1, 3, 9, 12} b) {red, orange, yellow, green, blue, indigo, purple} |
A set simply specifies the contents; the order is not important and it doesn’t matter if elements are repeated. A set is equal to another set if it has the exact same elements as another set. For example, set {3, 6, 8} is equal to set {6, 8, 3}. Also, note that repeated elements only need to be listed once. For example, the set represented by {1, 2, 3} is equal to the set {3, 1, 3, 2}.
We have to be careful when we define a set. For example, consider the set of the ten greatest movies of all time. Could you write this set down as a list of elements? What movies would you include? Would your friend pick different movies? The elements of this set are subjective. Since we cannot definitively define what elements are in the set of the greatest movies, this is an example of a statement that does not define a set because it is not well defined. A collection must be well defined in order to be considered a set. There cannot be any confusion or debate regarding what elements are in the set. The collection of the ten greatest movies of all time is not a set.
| Example 2 |
|
Some examples of collections that are not well-defined sets: a) The collection of the best pizza restaurants. (who decides? Your favorite restaurants or my favorite?) b) The collection of gas stations within ten miles of Mesa Community College. (which campus? Or does it mean any campus?) |
|
Some examples of collections that are well defined sets: a) The collection of the best pizza restaurants for 2018 selected by the Phoenix New Times. b) The collection of gas stations within ten miles of the Southern and Dobson campus of Mesa Community College. |
| You Try SV.1.A |
|
Let P be the collection of the best U.S. Presidents. Is P a set? |
| Notation |
|
We will use a variable to represent a set, to make it easier to refer to that set later. The symbol ∊ means “is an element of”. A set that contains no elements, { }, is called the empty set and can also be written ∅. Caution: ∅ is not the zero symbol, “0”. It is an abbreviation for the empty set. |
| You Try SV.1.B |
|
Let P be the set of living pigs that have wings and can fly. Is P well defined? What notation can be used for P? |
| Example 3 |
|
Let A = {1, 2, 3, 4} To notate that 2 is an element of the set, we write 2 ∊ A |
Some sets cannot be written by listing all the elements. This is because they have an infinite number of elements. For example, the set of even natural numbers cannot be listed. It is an infinite set. Infinite sets cannot not be fully written by listing all the elements. A set where we can write all the elements down is called a finite set. Note that sets can be written in list form using ellipsis, “…”, if there is a clear pattern for the reader to understand.
| Example 4 |
|
Some examples of sets that are infinite: a) The set of all odd natural numbers. We cannot write all the elements in list form, but we can write the set in list form using ellipsis, {1, 3, 5, 7, …}. b) The set of all real numbers greater than zero and less than ten. Hint: Real numbers |
|
Some examples of sets that are finite: a) The set of all students currently enrolled at Mesa Community College for the Fall 2019 semester. Although this set is very large, it can still be written down as a very long list. b) The set of natural numbers greater than zero and less than ten. This set can be written as {1, 2, 3, 4, 5, 6, 7, 8, 9} |
Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.
| Subset |
|
A subset of a set A is another set that contains some or all of the elements from set A. If B is a subset of A, we write B ⊆ A B is a subset of A can also be written as A ⊇ B Set A can equal set B for subsets The empty set is subset of all sets. A proper subset of a set A is another set that contains some but not all of the elements from set A – it contains fewer elements. If B is a proper subset of A, we write B ⊂ A If B is a proper subset of A can also be written as A ⊃ B Note: Set B ≠ Set A for proper subsets |
| Example 5 |
|
Consider these three sets A = the set of all even numbers B = {2, 4, 6} C = {2, 3, 4, 6} D = {2, 3, 4, 6} Here B is a proper subset of A, noted as B ⊂ A since every element of B is also an even number, so is an element of A. B is also a subset of A noted as B ⊆ A. It is also true that B ⊂ C. C is not a subset of A, since C contains an element, 3, that is not contained in A. Subsets can be equal to each other, so D is a subset of C, noted as D ⊆ C, but D is not a proper subset of C. |
| Example 6 |
|
Consider the following set of plays: {“Much Ado About Nothing”, “MacBeth”, “A Midsummer’s Night Dream”}. What is a larger set this might be a subset of? |
|
There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature. |
| You Try SV.1.C |
|
The set A = {1, 3, 5}. What is a larger set that this might be a subset of? |
Subsets are often used in the real world to list all the possible options in a situation where someone is given choices. Let’s say you are running a hotdog business. You offer your customers a choice of ketchup, mustard, and relish for their condiments. What are all the possible condiment choices for the hotdog orders?
The orders can be listed as all the possible subsets of the {ketchup, mustard, relish}. The possible options are: {ketchup, mustard, relish},
{ketchup, mustard},
{ketchup, relish},
{mustard, relish},
{ketchup},
{mustard},
{relish},
{ }
In this case the empty set corresponds to someone ordering a plain hotdog, no condiments.
Listing out all the subsets and proper subsets for a set can be time consuming. There is an easy way to figure out the number of subsets and proper subsets based on the number of elements in a set.
For 3 possible condiments, there are 8 possible subsets. Notice that 8 is equal 2 raised to the 3 or 23. The number of proper subsets is all the possible orders except for the subset using all the condiments, which is {ketchup, mustard, relish}. This means there are 7 proper subsets.
| Number of Subsets and Proper Subsets |
|
If the number of elements in a set is N, then the number of subsets is 2N and the number of proper subsets is 2N-1. Where N = the number of elements in the set |
| You Try SV.1.D |
|
The set A = {1, 3, 5}. What are the possible proper subsets of A? |
| You Try SV.1.E |
|
A pizza shop offers the following set of toppings: {pepperoni, sausage, Italian sausage, black olives, ham, mushrooms, bell peppers, pineapple}. How many different subsets of pizza toppings are possible? |
Section SV.1 – Answers to You Try Problems
SV.1.A
This set is not well defined. Do we mean a top 10 or the one best President? Just imagine the arguments that would happen if we all tried to agree which President was the best.
SV.1.B
The set is well defined. We can all agree that there are not living pigs with wings that can fly. We can say the set P is empty, P={ } or P=∅.
SV.1.C
There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The set of all integers. The set of all real numbers.
SV.1.D
{1, 3}, {1, 5}, {3, 5}, {1}, {3}, {5}, { }. Note there are 7 = 23 – 1 proper subsets.
SV.1.E
There are 256 = 28 different possible pizza orders since there are 8 different toppings.
