Name____________________________
Set Theory Tutorial
Definition: A set is simply a collection of object called members or elements.
A set may consist of numbers, such as the set of positive, odd integers: {1, 3, 5, 7, … } or objects such as the
days of the week: {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.
Let’s call the set of positive, odd integers set P. Then for an element in P we can represent its inclusion in
the set with the symbol ∈, which is read, “is an element of.”
So it is true that 1 ∈ 𝑃 and 3 ∈ 𝑃, etc.
Furthermore, it is true that a number such as 2 is not in the set, or 2 𝑃.
In general, there are 3 ways to represent a set.
1. we can use words:
2. we can make a list in roster notation:
3. we can use set-builder notation:
e.g. N is the set of natural numbers or counting numbers.
e.g. N = {1, 2, 3, …}
e.g. N = {x | x  N}, read “the set of all x such
that x is a natural number”
Further examples of set-builder notation:
(a) S, the set of awesome rock bands*:
𝑆 = {𝑚 | 𝑚 is an awesome rock band}
(b) The number 3 is a positive integer:
3 ∈ {𝑥 |𝑥 is a positive integer}
*by the way, Led Zeppelin ∈ 𝑆, Pink Floyd ∈ 𝑆, to name a few, but Kiss  S
The empty set (denoted ∅) has no elements but is still considered a set.
Example: The set of teachers under 20 years of age is an empty set.
Common number sets are listed below as reference:
ℝ
ℚ
ℕ
ℤ
ℤ+ , ℤ−
set of real numbers
a
set of rational numbers (can be written )
b
set of Natural numbers (sometimes includes zero; we will use {1,2,3,…}
set of integers
positive/negative integers, etc.
A finite set has a limited number of elements.
e.g. The set of students in this class
An infinite set has an unlimited number of elements.
e.g. The set of real numbers
The universe of a set or group of sets is the domain of allowable objects in a given situation.
Any object in the universe (or “universe of discourse”) is either an element of a set or it is not. For example, to consider the
set of students in this class who are juniors verses the set of students who are seniors, the universe would be all students in
this class. Note that this example excludes freshman and sophomores from any set but they are still part of the universe.
Usually the universe is clear from context, but sometimes it is helpful to define it explicitly. For instance, if the universe is the
set of students in a class, then the students with blonde hair could be written {𝑥 ∈ 𝑈 | a student 𝑥 has blonde hair} (read, “the
set of all students x in this class, such that each student has blonde hair”)
Further example:
Consider the statement 𝑥 2 < 9. If the universe of discourse for this statement were the set of all real numbers, then we might
represent the set of solutions as {𝑥 ∈ ℝ | 𝑥 2 < 9}, or in other words, the set of all real numbers between -3 and 3.
However, if the universe were the set of all integers, then the solution set could be represented as 𝑥 ∈ ℤ | 𝑥 2 < 9} =
{−2, −1, 0, 1, 2}
Thus, for example, 1.58 ∈ {𝑥 ∈ ℝ | 𝑥 2 < 9} but 1.58 ∉ {𝑥 ∈
make a large difference.
ℤ | 𝑥 2 < 9} and so it should be clear that the choice of universe can
Exercise 1: Represent each set below by using roster notation.
(a) The set S, where S is all positive integers that are divisible by 4
(b) The set P, where P is all odd perfect squares
(c) The set C, where C is all United States that begin with the letter A
(d) The set W, where W is the set of all even prime numbers
Exercise 2: Use set-builder notation to represent each set below:
(a) 𝑆, where 𝑆 is the set of all real numbers between -2 and 2.
(b) 𝐵, where 𝐵 is the set of whole numbers greater than 5
(c) M, the set of countries that begin with the letter M.
Definition: A conjunction () is formed by joining two statements with the word “and.”
Let A be the set of months with 30 days and let B be the set of months that end in “ember.”
Then 𝐴  𝐵 = {September, November}
Definition: A disjunction () is formed by joining two statements with the word “or.”
Let 𝐻 = {𝑥|𝑥 is a person who ate steak at the wedding} and
𝐺 = {𝑦|𝑦 is a person who ate salmon at the wedding}.
Then 𝐻 𝐺 is the number of people who ate either steak or salmon
Venn Diagrams:
__________________________________________
One way to represent or visualize sets is to use Venn diagrams:
Here we will build an example to introduce the next few terms.
Suppose the universe of discourse is all students enrolled
in a math class this year.
Let M be the set of students enrolled in Multivariable Calculus/Linear Algebra and let S represent the set
of all students enrolled in Statistics.
Definition: For any two sets S and T, we call the set consisting of the elements belonging to both S and T,
the intersection of S and T (in symbols, 𝑆 ∩ 𝑇 ). An intersection is also a conjunction.
Thus, 𝑆 ∩ 𝑇 = {𝑒 | 𝑒 ∈ 𝑆 𝐚𝐧𝐝 𝑒 ∈ 𝑇}
Example:
Let 𝑈 = {1, 3, 5, 7} and 𝑉 = {7, 6, 5, 4}.
Then 𝑈 ∩ 𝑉 = {5, 7}
𝑀 ∩ 𝑆 = the set of all students taking both
Multivariable Calculus and Statistics.
Definition: For any two sets S and T, we call the set consisting of the elements bellowing to at least one
of the given sets, the union of S and T (in symbols, 𝑆 ∪ 𝑇 ). A union is also a disjunction.
Thus, 𝑆 ∪ 𝑇 = {𝑒 | 𝑒 ∈ 𝑆 𝐨𝐫 𝑒 ∈ 𝑇}
Example:
Let 𝐶 = {0, 1, 2,3} 𝑎𝑛𝑑 𝐷 = {1, 3, 5}.
Then 𝐶 ∪ 𝐷 = {0, 1, 2, 3, 5}
𝑀 ∪ 𝑆 = the set of all students taking either
Multivariable Calculus or Statistics.
Definition: the complement of a set consists of all objects not in the set, usually denoted 𝐴′ or ¬𝐴
If A is the set of clothes that I wore this week, then ¬𝐴 represents all the clothes I did not wear.
Let C be the set of all students enrolled in a math class
this year, but who are not taking multivariable calculus
or statistics. Then 𝐶 = (𝑀 ∪ 𝑆)′ or 𝐶 = ¬(𝑀 ∪ 𝑆)
Definition: the difference between sets, denoted 𝐴\𝐵
Let 𝐴 = {1, 3, 5, 6, 9} and 𝐵 = {1, 5, 6}. Then 𝐴\𝐵 = {3, 9}.
𝑀\𝑆, for example, is the set of students taking
multivariable calculus but who are not taking statistics.
Exercise 3: Redraw the diagram below and shade the regions representing each of the following:
(a) 𝑆 ∩ 𝑇
(e) (𝑇\𝑆) ∪ 𝑅
(b) 𝑆 ∪ 𝑇
(f) (𝑇\𝑆) ∪ (𝑇 ∩ 𝑆)
(c) 𝑆 ∪ 𝑅
(d) 𝑆 ∩ 𝑅
Exercise 4: Suppose 𝐴 = {1, 2, 3, 4, 5} and 𝐵 = {2, 4, 6, 8, 10}. List the elements of the following sets:
1. 𝐴 ∩ 𝐵
2. 𝐴 ∪ 𝐵
3. 𝐴\𝐵
4. (𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵)
5. (𝐴\𝐵) ∪ (𝐵\𝐴)
Exercise 5: Redraw the diagram below and shade the regions representing each of the following:
(𝑎) 𝐴 ∩ 𝐶
(𝑏) 𝐴 ∪ (𝐵\𝐶)
(𝑐) ¬[(𝐵 ∪ 𝐶)\𝐴]
(𝑑) (𝐴\𝐶) ∪ (𝐵 ∩ 𝐶)
Exercise 6: Let P stand for the statement “I will buy the pants” and S for the statement “I will buy the
shirt.” What English sentences are represented by the following expressions?
(𝑎) ¬(𝑃  ¬𝑆)
(𝑏) ¬𝑃  ¬𝑆)
(𝑐) ¬𝑃  ¬𝑆
Exercise 7: Let S stand for the statement “Steve is happy,” and G for “George is happy.” What English
sentences are represented by the following expressions?
(a) (𝑆  𝐺)  (¬𝑆  ¬𝐺)
(b) [𝑆  (𝐺¬𝑆)]  ¬𝐺
(c) 𝑆  [G  (¬𝑆  ¬𝐺)]
Exercise 8: Let the universe U be the members of the United Federation of Planets’ Star Fleet. Assume
that members are either terrestrial (Earth-men) or Vulcan. Let V be the set of Vulcan members of Star
Fleet. Let A be the set of Star Fleet members who graduated from the Academy. Let O be the set of Star
Fleet Officers. Let K be the set of Star Fleet members whose last names are “Kirk.”
Translate the following descriptions into mathematical set notation:
(a) Vulcans whose last names are “Kirk.”
(b) Officers who graduated from the Academy.
(c) Enlisted members (non-officers) whose last name is not “Kirk.”
(d) Humans who did not graduate from the academy but became officers anyway.
(e) Members who are either officers or Vulcans, but graduated from the academy.
Subsets:_________________________________________________
If every element of a set A is also an element of the set B, we say that A is a subset of B and write 𝐴  𝐵.
If 𝐴  𝐵 and 𝐵  𝐴, then the sets contain the same elements and we can say that A and B are equal, or
𝐴 = 𝐵.
If 𝐴  𝐵 but 𝐴 ≠ 𝐵, then 𝐴 is called a proper subset of 𝐵 and we write 𝐴 ⊂ 𝐵.
Example:
𝐴 = {𝑎, 𝑏, 𝑐, 𝑑},
𝐵 = {𝑎, 𝑏, 𝑐}
𝐶 = {𝑐, 𝑑, 𝑎, 𝑏}
Since every element of B is also an element of A, then 𝐵  𝐴. In fact, B is a proper subset of A since A is
not a subset of B.
Since 𝐴  𝐶 and 𝐶  𝐴 , then 𝐴 = 𝐶.
Glossary of Symbols
Symbol
−, ¬, 𝑜𝑟 ~


∩
∪
→
↔
∀

E!
{}
…
{𝑥| ___________}
∈


∅
𝐴∪𝐵
𝐴∩𝐵
𝐴\𝐵
ℝ
ℚ
ℕ
ℤ
ℤ+ , ℤ−
F
P(A)
Name of Symbol
Negation (“is not”)
Conjunction
Disjunction
Intersection
Union
Implies
Equivalence (biconditional)
Universal Quantifier (“for all/every”)
Existential quantifier (“there exists/there is”)
Uniqueness quantifier (“there exists a unique…)
Set roster notation
and so on; continues indefinitely
set builder notation
is an element of
is not an element of
is a subset of
Empty set
Union
Intersection
Difference
set of real numbers
a
set of rational numbers (can be written b)
set of Natural numbers (sometimes includes zero;
we will use {1,2,3,…}
set of integers
positive/negative integers, etc.
family of sets
power set of A
Suppes
Exercise 3 solutions
(a)
(b)
(c)
(d)
𝑆∩𝑅 =∅
extras
U
S
M
U
M
S