Download Sets, Venn Diagrams, and Set Operations

Section 5.1 – Sets, Venn Diagrams, and Set Operations
If we think about a set of dinnerware, a set of tires or a set of golf clubs, we could describe
exactly what is included in the set. In the set of dinnerware, for example, there could be plates,
bowls, cups, and saucers, but not other items such as books and cell phones. Books and cell
phones are not what we mean when we speak of “dinnerware.” Similarly, if we consider a set of
golf clubs, it would consist of woods, drivers, irons and a putter, but not towels or lamps.
The same is true in mathematics. A set is a collection of objects. We have a good idea of the
meaning of a set, just by referring to the concept in everyday life. This term is one of a handful
of “undefined” terms in mathematics, meaning that we do not use other mathematical words to
describe it. An undefined term is a basic concept on which other definitions are based.
In our discussion of sets in this course, we are interested in well-defined sets. That is, when
described, we can tell immediately if an item is in the set or is not in the set.
Set Terminology
We typically refer to sets by using capital letters of the English alphabet. The items that are in a
set are called elements. We can describe a set by enclosing the elements of the set inside
brackets.
Some examples of set notation are shown below:
D = {knife, fork, spoon}
H = {Bob, Sanjay, Carol, Tim}
V = {3, 5, 7, 9, 11}
There are two methods for describing sets, the roster method (list method) and set-builder
notation (rule method). Sets D, H, V and W above are all written using the roster method. When
using the roster method, we list all of the items that are in the set.
The elements of a set should be distinct. .
Example 1: Let W represent the set of letters in the word MISSISSIPPI. What is W ?
Solution: Each element of a set should be distinct from the others. We therefore only list each
letter once, so W = {M, I, S, P} .
***
Consider the set B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} . We can instead write set B in a type of notation
known as set-builder notation. Set-builder notation describes the elements of a set using a rule.
We can write B = { x x is a counting number less than 11} , which is read “B equals the set of all
x such that x is a counting number less than 11.”
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Section 5.1
B=
{
x
|
↑
The set of
↑
all x
↑
such that
x is a counting number less than 11}
↑
x is a counting number less than 11
The “such that” symbol can be written as either a vertical bar or a colon. Therefore, we can
alternatively write: B = { x : x is a counting number less than 11} . In this text, we use the vertical
bar to denote the “such that” symbol.
We use the notation a ∈ A to indicate that the element a is an element of set A. We use the
notation b ∉ A to indicate that an element is not in a set.
So, using the set B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , we can write 5 ∈ B to mean “5 is an element of
set B.” We can write 20 ∉ B to mean “20 is not an element of set B.” Set B is well-defined, since
we can conclusively determine whether or not an element is a member of its set.
We say that two sets are equal sets if they contain exactly the same elements. Given the sets
M = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48} and
N = { x x is an even number between 1 and 49} , we can see that M = N .
The order of elements does not matter in set notation. If Q = {2, 4, 6} and R = {4, 6, 2} , then
Q = R.
A set that contains no elements is called the empty set and is noted by ∅ , or is alternatively
written as
{}
.
Set A is a subset of set B if every element that is in set A is also in set B. The notation for this is
A ⊆ B and is read “A is a subset of B.” This is sometimes read as “A is contained in B.”
Set A is a proper subset of set B if every element that is in set A is also in set B and there is at
least one element in set B that is not in set A. The notation for this is A ⊂ B and is read “A is a
proper subset of B.” (Notice that the proper subset symbol looks a lot like a “less than” symbol.
Although they do not have the exact same meaning, this similarity may help us to remember that
when A ⊂ B , all elements of A are also in B, and the number of elements in A is less than the
number of elements in B.)
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Section 5.1
Example 2: Suppose that A = {0, 2, 4, 6, 8, 10} , B = {2, 8, 10} , C = {0, 2, 4, 6, 8, 10} , and
D = {0, 1, 2, 3} . Identify each statement below as true or false.
A. 3 ∈ A
E.
D⊆C
B.
B⊆A
F.
5∉ D
C.
B⊂A
G.
A⊆C
D.
A⊆ B
H. C ⊄ A
Solution:
A.
The statement 3 ∈ A is false because 3 is not an element of A = {0, 2, 4, 6, 8, 10} .
B.
The statement B ⊆ A is true. Every element in B = {2, 8, 10} is also in
A = {0, 2, 4, 6, 8, 10} . Therefore, B is a subset of A.
C.
The statement B ⊂ A is true. Every element in B = {2, 8, 10} is also in
A = {0, 2, 4, 6, 8, 10} , and there is at least one element in set A that is not in set B.
Therefore, B is a proper subset of A.
D.
The statement A ⊆ B is false. The numbers 0, 4, and 6 are in A = {0, 2, 4, 6, 8, 10} and
none of them are in B = {2, 8, 10} .
E.
The statement is D ⊆ C false. The numbers 1 and 3 are in D = {0, 1, 2, 3} and neither of
them is in C = {0, 2, 4, 6, 8, 10} .
F.
The statement 5 ∉ D is true. The number 5 is not an element of D = {0, 1, 2, 3} .
G.
The statement A ⊆ C is true. Every element in A = {0, 2, 4, 6, 8, 10} is also in set
C = {0, 2, 4, 6, 8, 10} . Since A = C , we note that a set is a subset of itself.
H.
The statement C ⊄ A is true. The statement C ⊄ A means “C is not a proper subset of
A.” Every element in set C = {0, 2, 4, 6, 8, 10} is also in set A = {0, 2, 4, 6, 8, 10} , and
the two sets are equal. For set C to be a proper subset of set A, there needs to be at least
one element in set A that is not in set C, which is not the case. Therefore, C is not a
proper subset of A.
***
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Section 5.1
Example 3: Let C = {1, 2, 3} .
A. List all subsets of C.
B. List all proper subsets of C.
Solution:
A.
There is one 3-element subset: {1, 2, 3} .
(Remember that the order in which elements are written does not matter, so {1, 2, 3} ,
{2, 1, 3} and {3, 2, 1} all represent the same set and it should only be listed once.)
There are three 2-element subsets: {1, 2} , {2, 3} , and {1, 3} .
There are three 1-element subsets: {1} , {2} , and {3} .
There is one 0-element set: ∅ .
Therefore, the subsets of C are {1, 2, 3} , {1, 2} , {2, 3} , {1, 3} , {1} , {2} , {3} , and ∅ .
B.
Next, we want to list all proper subsets of C = {1, 2, 3} . For a subset to be a proper
subset, there must be at least one element in C that is not in the listed subset. This means
that the set {1, 2, 3} is not a proper subset of itself. However, all of the other subsets
listed above meet the definition.
Therefore, the proper subsets of C are {1, 2} , {2, 3} , {1, 3} , {1} , {2} , {3} , and ∅ .
***
There is a formula to determine the number of subsets which are contained in a set.
The Number of Subsets in a Set
If a set contains n distinct elements, then the set has 2n subsets.
If we want to find the number of proper subsets of a set, we take the number of subsets and
subtract 1, since the original set is not a proper subset of itself.
The Number of Proper Subsets in a Set
If a set contains n distinct elements, then there are 2 n − 1 proper subsets.
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Section 5.1
We can verify that the above formulas work for Example 3. Set C contained three elements, so
n = 3 . Using the above formulas, the number of subsets should be 23 and the number of proper
subsets should be 23 − 1 . If we count the number of subsets and proper subsets which we
obtained, we can see that we found 23 = 8 subsets, and 23 − 1 = 7 proper subsets.
Example 4: Suppose that set D has 4 distinct elements.
A. Find the number of subsets of D.
B. Find the number of proper subsets of D.
Solution: Set D has 4 elements, so n = 4 .
A.
The formula for the number of subsets is 2n . Since 2 4 = 16 , D has 16 subsets.
B.
The formula for the number of proper subsets is 2 n − 1 . Since 24 − 1 = 15 , D has 15
proper subsets.
***
Venn Diagrams
A Venn diagram is a very useful tool for showing the relationships between sets. Venn
diagrams consist of a rectangle with one or more shapes (usually circles) inside the rectangle.
The rectangle represents all of the elements that we are interested in for a given situation. This
set is called the universal set and is noted by U.
The relationship A ⊆ U is shown by the following Venn diagram:
Suppose that A ⊆ U , B ⊆ U , and A = B . The Venn diagram below represents this relationship.
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Section 5.1
The following Venn diagram illustrates that A ⊆ U , B ⊆ U , and A ⊆ B . Notice that all points
in set A are also in set B.
There are two different Venn diagrams which represent the relationship A ⊄ B . In the diagram
shown below at the left, the two sets overlap, but there is a region of set A that is not contained in
set B. In the diagram shown below at the right, the two sets do not overlap at all, so none of set A
is contained in set B. In both situations, A is not a subset of B.
Set Operations
We will use three set operations to form new sets from given sets. The first set operation we will
discuss is called the union of sets. We will start with a universal set U along with two sets A and
B that are subsets of U.
Union of Sets
The union of sets A and B, which is written as A ∪ B , is the set of all elements
that belong to either A or B (or both). In set builder notation,
A ∪ B = { x x ∈ A or x ∈ B (or both)}
In simpler terms, the union of two sets is the set of distinct objects that exist when the sets are
combined.- (Be sure to list each shared element only once.)
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Section 5.1
Notice in the definition that the union of two sets consists of all elements that belong either to the
first set, or to the second set, or both. In mathematical terms, we can leave the “or both” part of
the definition out, because the word “or” in the expression “ x ∈ A or x ∈ B ” automatically
implies that the shared elements are included. Mathematically speaking, if Nancy says that she is
having a hamburger or a hot dog for dinner, this means that she is having a hamburger, a hot dog,
or both. In everyday life, however, we often use the word “or” in cases where we select one or
the other, e.g. either a hamburger or a hot dog, but not both. To avoid confusion, the “or both” is
included in the definition even though it is mathematically extraneous, as a reminder to the
reader that shared elements are always included in the union of the sets.
In the Venn diagram shown below, A ∪ B is represented by the shaded region. Any element in
the shaded region is either in set A, or set B, or both.
A∪ B
Be careful not to get the union symbol, ∪ , confused with the universal set, U.
The next set operation we will discuss is called the intersection of sets.
Intersection of Sets
The intersection of sets A and B, which is written as A ∩ B , is the set of all
elements that belong to both A and B. In set-builder notation,
A ∩ B = { x x ∈ A and x ∈ B}
In the Venn diagram shown below, A ∩ B is represented by the shaded region. Any element in
the shaded region belongs to both set A and set B. The intersection represents the overlap of the
two sets.
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Section 5.1
A∩ B
If there are no elements in the intersection of two sets, then the sets are called disjoint sets.
Using set notation, two sets are disjoint if A ∩ B = ∅ .
In the following Venn diagram, A and B are disjoint, since they do not intersect.
The last basic set operation we will discuss is called the complement of a set.
Complement of a Set
The complement of A, which is written as A c , is the set of all elements that are in
the universal set U but are not in set A. In set-builder notation,
{x
x ∈ U , x ∉ A}
In the Venn diagram shown below, A c is the shaded region.
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Section 5.1
Ac
A c is also sometimes written as A′ or A . (We will only use the A c notation in this text.)
Example 5: Suppose that U = {1, 2, 3, … , 10} , A = {1, 3, 4, 6} , B = {2, 4, 6, 8, 10} , and
D = {2, 5, 7, 8, 10} . Find each of the following:
A.
A∩ B
B.
B∪D
C.
Dc
D.
Ac ∪ B
E.
Bc ∩ D
Solution:
A.
The intersection, A ∩ B , is the set of numbers that are contained both in A = {1, 3, 4, 6}
and in B = {2, 4, 6, 8, 10} . The numbers 4 and 6 are contained in both sets. Therefore,
A ∩ B = {4, 6} .
B.
The union, B ∪ D , is the set of all numbers that are in set B = {2, 4, 6, 8, 10} or set
D = {2, 5, 7, 8, 10} , or both. We combine the sets, being careful to list each shared
element only once. Therefore, B ∪ D = {2, 4, 5, 6, 7, 8, 10} .
C.
The complement of D is the set of all numbers in the universal set U = {1, 2, 3, … , 10}
that are not in set D = {2, 5, 7, 8, 10} . Therefore, D c = {1, 3, 4, 6, 9} .
D.
To find A c ∪ B , let us start by first finding A c . The complement of A is the set of all
numbers in the universal set U = {1, 2, 3, … , 10} that are not in the set A = {1, 3, 4, 6} .
Hence, A c = {2, 5, 7, 8, 9, 10} . To find A c ∪ B , we list all of the elements that are in
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Section 5.1
A c = {2, 5, 7, 8, 9, 10} or in B = {2, 4, 6, 8, 10} , or both. (Combine the sets, and be sure
to list each shared element only once.) We can conclude that
A c ∪ B = {2, 4, 5, 6, 7, 8, 9, 10} .
E.
To find B c ∩ D , let us start by first finding B c . The complement of B is the set of all
numbers in the universal set U = {1, 2, 3, … , 10} that are not in the set
B = {2, 4, 6, 8, 10} . Hence, B c = {1, 3, 5, 7, 9} . To find B c ∩ D , we list the elements
which are contained both in B c = {1, 3, 5, 7, 9} and in D = {2, 5, 7, 8, 10} . We can
conclude that B c ∩ D = {5, 7} .
***
Properties of Set Operations
Here are some additional properties involving set operations:
Properties Involving Set Complementation
Suppose that U is a universal set and A is a subset of U. Then each of the
following statements is true.
Uc =∅
∅ c =U
(A )
A∪ Ac = U
c c
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=A
A∩ Ac = ∅
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Section 5.1
Properties of Unions and Intersections
Suppose that U is a universal set and A, B, and C are all subsets of U.
Set unions and set intersections have the following properties, as indicated below:
Commutative:
A∪ B = B ∪ A
A∩ B = B ∩ A
Associative:
A ∪ ( B ∪ C ) = ( A ∪ B) ∪ C
A ∩ ( B ∩ C ) = ( A ∩ B) ∩ C
Distributive:
A ∪ ( B ∩ C ) = ( A ∪ B) ∩ ( A ∪ C )
A ∩ ( B ∪ C ) = ( A ∩ B) ∪ ( A ∩ C )
The following laws, known as DeMorgan’s Laws, often allow us to replace an expression
involving sets with a simpler equivalent expression.
DeMorgan’s Laws
Suppose that A and B are sets. Then the following statements are true:
( A ∪ B)
c
= Ac ∩ B c
( A ∩ B)
c
= Ac ∪ B c
Properties of set operations can extend to any finite number of sets.
All of the properties involving set operations can be verified using Venn diagrams. One
verification is shown below in Example 6.
Example 6: Use Venn diagrams to verify the following rule from DeMorgan’s Law:
( A ∪ B)
c
= Ac ∩ B c
Solution: We can draw Venn diagrams of both sides of the equation and show that the results are
the same.
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Section 5.1
For the left hand side of the equation, ( A ∪ B ) , we first shade A ∪ B and then find the
c
complement of that set.
A∪ B
( A ∪ B)
c
For the right hand side of the equation, A c ∩ B c , we first find A c and B c and then find the
regions that the two sets have in common.
Ac
Bc
Ac ∩ B c
Note that the Venn diagram for ( A ∪ B ) is identical to the Venn diagram for A c ∩ B c .
c
Therefore, ( A ∪ B ) = A c ∩ B c .
c
***
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Section 5.1
Example 7: Use a Venn diagram to illustrate A ∪ B c .
Solution: First, let us draw a Venn diagram to represent set A, and another Venn diagram to
represent set B c .
A
Bc
Since the problem calls for the union of these two sets, A ∪ B c , the solution consists of all
regions of set A along with all regions of set B c . Any region shaded in either Venn diagram (or
shared by both) is included in the final answer. The Venn diagram for A ∪ B c is shown below.
A∪ Bc
***
Example 8: Use a Venn diagram to illustrate A ∩ ( B ∪ C ) .
Solution: First, let us draw set A, and then draw set B ∪ C .
B ∪C
A
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Section 5.1
To find A ∩ ( B ∪ C ) , we draw another Venn diagram and shade the regions where the above
sets intersect (overlap).
A∩(B ∪C)
***
Example 9: Use a Venn diagram to illustrate A ∪ ( B c ∪ C ) .
c
Solution: ( B c ∪ C ) can be rewritten as B ∩ C c using DeMorgan’s Laws. This means that the
c
Venn diagram for A ∪ ( B c ∪ C ) is the same as the Venn diagram for A ∪ ( B ∩ C c ) . We will
c
focus on illustrating A ∪ ( B ∩ C c ) , since it has a simpler form than A ∪ ( B c ∪ C ) .
c
To illustrate A ∪ ( B ∩ C c ) , let us first focus on the operations inside the parentheses. B ∩ C c
represents the elements in set B that are not also in set C. The result is shown below.
B∩Cc
Finally, we want to find the union of B ∩ C c (shown above) with set A. Remember that by
DeMorgan’s Laws, A ∪ ( B c ∪ C ) = A ∪ ( B ∩ C c ) . The final result is shown below.
c
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Section 5.1
A∪(B ∩C c )
***
Example 10: Suppose that U is the set of all UH students, P is the set of all UH students who
belong to a fraternity/sorority, C is the set of all UH students who live on campus, and B is the
set of all UH students who are business majors. Describe each of the following sets using words:
A. C ∪ ( P ∩ B )
B.
P c ∩ (C ∪ B )
C.
B ∩C ∩ P
Solution:
A.
P ∩ B is the set of all business majors who are also members of fraternities/sororities.
The union of that set with set C is as follows:
The set of all UH students who either live on campus or who are both business majors
and belong to fraternities/sororities.
B.
The set C ∪ B is the set of all UH students who either live on campus or are business
majors or both. Set P c is the set of all UH students who do not belong to
fraternities/sororities. The intersection of these two sets is as follows:
The set of all UH students who are not members of fraternities/sororities, but who either
live on campus or are business majors.
C.
This set requires that students are in all three sets. This set is:
The set of all UH students who are business majors living on campus and are also
members of fraternities/sororities.
***
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Section 5.1
Example 11: Suppose that U is the set of all UH students, P is the set of all UH students who
belong to a fraternity/sorority, C is the set of all UH students who live on campus, and B is the
set of all UH students who are business majors. Write each of the following sets using set
operations.
A.
The set of all UH business majors who do not live on campus and are not members of
fraternities/sororities.
B.
The set of all UH students who live on campus and are members of fraternities/sororities
who major in something other than business.
C.
The set of all UH students who either live on campus or major in business but who are
not members of fraternities/sororities.
Solution:
A.
C c ∩ P c represents the students do not live on campus and are not members of
fraternities/sororities. The selected students are also business majors, so the final answer
is B ∩ ( C c ∩ P c ) . We can also use DeMorgan’s Law to write this as B ∩ ( C ∪ P ) .
c
B.
C ∩ P represents the students who live on campus and are members of
fraternities/sororities. The selected students are not business majors, so the final answer is
B c ∩ (C ∩ P ) .
C.
C ∪ B represents the students who either live on campus or major in business. However,
the selected students are not members of fraternities/sororities, so the final answer is
P c ∩ (C ∪ B ) .
***
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Section 5.1