Math 141: Business Mathematics I
Fall 2015
§6.1 Sets and Set Operations
Instructor: Yeong-Chyuan Chung
Outline
• Sets - Terminology and Notation
• Set Operations
Sets
A set is a well-defined collection of objects. Here, “well-defined” means that if we are given
an object, then we should be able to determine whether or not it belongs in the collection.
Some examples of sets:
• The collection of all people enrolled in this class whose last name starts with “A”.
• The collection of all people enrolled in this class who were born before the year 1995.
• The collection of all elephants that fly.
The objects in a set are called the elements of the set.
We usually use uppercase letters for sets, and lowercase letters for elements of sets.
There are two ways to write down a set symbolically:
• Roster notation - List all elements of the set, separated by commas, and enclosed within
a pair of braces. For example, {a, b, c} is the set consisting of the first three letters of
the English alphabet.
• Set-builder notation - Give a rule that describes the property or properties an object
must satisfy to be an element of the set. For example, {x|x is a letter of the English alphabet}.
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§6.1 Sets and Set Operations
If a is an element of the set A, we write a ∈ A (read “a belongs to A” or “a is an element
of A”). On the other hand, if a is not an element of A, then we write a ∈
/ A (read “a does
not belong to A” or “a is not an element of A”).
Two sets A and B are equal, written A = B, if and only if they have exactly the same
elements.
Example. Let A = {1, 2, 3}, B = {2, 1, 3}, C = {2, 4}. Which sets are equal?
If every element of a set A is also an element of another set B, then we say that A is a
subset of B, and write A ⊆ B.
Example. Let A = {1, 2}, B = {1, 2, 3}, C = {1, 2, 3, 4}. Which sets are subsets of others?
Example. If A ⊆ B, and B ⊆ A, then what can we deduce about the sets A and B?
If A is a subset of B, but A and B are not equal, then we say that A is a proper subset of
B, and write A ⊂ B. In other words, this means that every element of A is also an element
of B, but there is at least one element of B that is not an element of A.
Note: Be careful to distinguish between the notations ∈ and ⊆ (or ⊂). The first one, ∈, is
used to describe a relationship (membership) between an element and a set. The other one,
⊆ (or ⊂), is used to describe a relationship (containment) between two sets.
§6.1 Sets and Set Operations
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Example. If A = {1, 2}, determine whether each of the following is true or false.
(a) 1 ∈ A
(b) 1 ⊆ A
(c) {1} ∈ A
(d) {1} ⊆ A
(e) {1} ⊂ A
There is a special set, called the empty set, and it is the set that has no elements. This set
is denoted by ∅. Sometimes we also use {} to denote this set.
Note: {∅} is not the same as ∅ or {}.
The empty set is a subset of every set. (You need a rule of logic to see this.)
Example. List all the subsets of the set A = {a, b, c}. (How many should there be?)
A universal set is the set of all elements of interest in a particular context. It changes
according to the context.
We can use Venn diagrams to visually represent sets. The universal set U is represented
by a (large) rectangle, and we draw circles inside the rectangle for each subset of interest
such that the relationships between the subsets are represented appropriately.
Example. Use a Venn diagram to illustrate the following statement: The set A is a proper
subset of the set B.
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§6.1 Sets and Set Operations
Set Operations
There are three basic operations that we can perform on sets to get new sets.
The union of two sets A and B is the set of all elements that belong to either A or B (or
both). This set is denoted by A ∪ B. It is illustrated in a Venn diagram as follows:
The intersection of two sets A and B is the set of all elements that belong to both A and
B. This set is denoted by A ∩ B. It is illustrated in a Venn diagram as follows:
If the intersection of two sets A and B is empty, i.e. A ∩ B = ∅, then we say that A and B
are disjoint.
If U is a universal set, and A is a subset of U , then the complement of A is the set of all
elements that belong to U but do not belong to A. This set is denoted by Ac . It is illustrated
in a Venn diagram as follows:
§6.1 Sets and Set Operations
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Example. Let U denote the set of all students in the business college of a certain university.
Let
A = {x ∈ U |x had taken a course in accounting}
B = {x ∈ U |x had taken a course in economics}
C = {x ∈ U |x had taken a course in marketing}
Describe the following sets in words:
(a) Ac
(b) B ∪ C
(c) A ∩ B c
Use set notation to represent the following sets (in terms of A, B, and C):
(a) The set of students who have had at least one of the three courses.
(b) The set of students who have had courses in accounting and economics but not marketing.
(c) The set of students who have had courses in economics but not courses in accounting
or marketing.
(d) The set of students who have had exactly one of the three courses.
§6.1 Sets and Set Operations
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Rules
Complementation: If U is a universal set, and A is a subset of U , then
• Uc = ∅
• ∅c = U
• (Ac )c = A
• A ∪ Ac = U
• A ∩ Ac = ∅
Unions and intersections: Let U be a universal set, and let A, B, C be subsets of U . Then
• A∪B =B∪A
• A∩B =B∩A
• A ∪ (B ∪ C) = (A ∪ B) ∪ C
• A ∩ (B ∩ C) = (A ∩ B) ∩ C
• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan’s laws:
• (A ∪ B)c = Ac ∩ B c
• (A ∩ B)c = Ac ∪ B c
Exercise: Verify these rules using Venn diagrams.