Notes for math 141
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1.1
Section 6.1-6.3
Finite Mathematics
6.1 Set and Set Operations
Set Terminology and Notation
Definitions: A set is a well-defined collection of objects usually denoted by uppercase letters.
The elements, or members, of a set are denoted by lowercase letters.
Set Notations:
1. Roster Notation: List of each element between braces
Example 1:
2. Set − builder Notation: A rule is given that describes the property an object x
must satisfy to qualify for membership in the set.
Example 2:
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Notation: If a is an element of a set A, we write a ∈ A. If a doesnt belong to A we
write ∈/ A
Example 3: Let A = {1, 2, 3}.
Definition: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. (Note: The elements do NOT have to be in the same order.)
Example 4: LetA = {1, 2, 3}, A = {2, 1, 3}, C = {1, 2, 3, 4}
Definition ∶ If every element of a set A is also an element of a set B, then we say
that A is a subset of B and write A ⊆ B.
Definition: A is a proper subset of B if A is a subset of B but A does not equal B. (i.e. We write A ⊂ B if A ⊆ B and there exists at least one element in B that
is not in A)
Example 5: Let A = {1, 2, 3}, A = {2, 1, 3}, C = {1, 2, 3, 4}
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Definition: The set that contains no elements is called the empty set and is denoted by ∅ (not {∅}). The empty set is a subset of every set.
Example 6: List all subsets of A = {1, 2, 3}
Definition: A universal set, U, is the set of all elements of interest in a particular matter. We have different universal sets for different problems.
We use Venn Diagrams to visually represent sets. The Universal Set U is denoted
by a rectangle. Subsets of U are represented by circles inside the rectangle.
Example 7 Venn Diagrams
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Section 6.1-6.3
Notes for math 141
1.2
Finite Mathematics
Set operations
Definitions: Given a universal set U and a set A ⊆ U , the complement of A,
written as Ac is all elements that are contained in the universal set U , but not in A,
that is
Ac = {x∣x ∈ U, x ∉ A}
Example 8: Let U = {1, 2, 3, 4, a, b, c}, A = {1, 2, a} and B = {x∣x is an even number in U }.
Complement Rules: If U is a universal set, we must always have
U c = ∅,
If A is any subset of U , then
∅c = U
(Ac )c = A
Definition: The union of two sets A and B, written A∪B, is the set of all elements
that belong to A and B, or to both. Thus
A ∪ B = {x∣x ∈ A or x ∈ B or both}
Example 9: Let U = {1, 2, 3, 4, a, b, c, d}, A = {1, 2, a, b}, B = {2, 3, a, b, c} and
C = {2, 4, a, c}. Find A ∪ B, A ∪ Ac , A ∪ B ∪ C and A ∪ (B ∪ C).
For set union, we have the following properties
A ∪ B = B ∪ A,
A ∪ Ac = U,
and
A ∪ B ∪ C = A ∪ (B ∪ C)
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Definition: The intersection of two sets A and B, written as A ∩ B, is the set of
all elements that belong to both the set A and to the set B. Thus,
A ∩ B = {x∣x ∈ A and x ∈ B}
Example 10: Let U = {1, 2, 3, 4, a, b, c, d}, A = {1, 2, a, b}, B = {2, 3, a, b, c}, C =
{2, 4, a, c} and D = {3, c, d}. Find A ∩ B, A ∩ Ac , A ∩ D, A ∩ B ∩ C and A ∩ (B ∩ C).
Definition: Two sets A and B are disjoint if they have no elements in common,
that is , if A ∩ B = ∅.
For set union, we have the following properties
A ∩ B = B ∩ A,
A ∩ Ac = ∅,
and A ∩ B ∩ C = A ∩ (B ∩ C)
Example 11: Let A = {1, 3, 5, 7}, B = {2, 4, 6, 8}. Are A and B disjoint?
De Morgan Law
(A ∪ B)c = Ac ∩ B c
(A ∩ B)c = Ac ∪ B c
Example 12: Lets prove DeMorgans Law using Venn Diagrams
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Distributive laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Example 12: Shade the sets A ∩ (B ∩ C) , A ∩ (B ∪ C) and A ∩ (B ∪ C)c in the Venn
diagram below.
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 13: Let U be the universal set consisting of all the students attending the
class Math 141-508, and
A = {x∣x likes jazz music}
B = {x∣x likes rock music}
C = {x∣x likes classical music}
Describe the following sets in words:
a. A ∩ B c
b. (A ∪ B) ∩ C c
c. A ∩ (B ∪ C)c
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Section 6.1-6.3
Notes for math 141
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Finite Mathematics
6.2: The Number of Elements in a Set
If A is a set with a finite number of elements, we denote the number of elements in
A by n(A). n(A) is also called cardinality of the set A. By the definition of empty
set, n(∅) = 0.
Example 1: IfA = {x∣x is a letter in the English Alphabet}, B= {a, b, c}, and
C = ∅, find n(A), n(B) and n(C).
The Number in the Union of Two Sets
For any finite sets A and B,
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
If A and B are disjoint sets , then
n(A ∪ B) = n(A) + n(B)
How to understand this from Venn Diagrams?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 2: In a survey of 100 students, 30 can speak French, 40 can speak Spanish
and 10 can speak both French and Spanish.
• (a) How many student can speak French but not Spanish?
• (b) How many student can speak French or Spanish?
• (c) How many student can speak neither French nor Spanish?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
The Number in the Union of Three Sets
n(A ∪ B ∪ C = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + (A ∩ B ∩ C)
How to understand this from Venn Diagrams?
Example 3: If n(A) = 12, n(B) = 12, n(A ∩ B) = 5, n(A ∩ C) = 5, n(B ∩ C) =
4, n(A ∩ B ∩ C) = 2, andn(A ∪ B ∪ C) = 25, find n(C).
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 4: In a survey of 100 students, 30 can speak French, 40 can speak Spanish,
25 can speak Russian, 10 can speak both French and Spanish, 7 can speak Spanish and
Russian, 5 can speak French and Russian and 2 can speak all these three languages.
Let A, B and C denote the set of all the students speaking French, Spanish and
Russian, respectively.
1. Use the survey results above to find the number of students in each section of
the given Venn diagram
2. How many students can speak Spanish but not Russian?
3. How many students can speak exactly one of the three languages?
4. How many students can speak none of the three languages?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 5: To help plan the number of meals to be prepared in a college cafeteria,
a survey was conducted, and the following data were obtained:
8 students ate only breakfast.
80 students ate only lunch.
96 students ate exactly 2 meals.
68 students ate breakfast and lunch.
58 students ate all three meals.
100 students did not eat dinner.
112 students ate breakfast and dinner.
99 students ate exactly 1 meal.
How many students were surveyed?
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Notes for math 141
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Section 6.1-6.3
Finite Mathematics
Section 6.3: The multiplication principle
Example 1: A yogurt shop has 3 flavors (chocolate, vanilla, strawberry) and 2 sizes
( medium, and large). How many different yogurts can be ordered?
Multiplication Principle: Suppose a task T1 can be performed in N1 ways, a
task T2 can be performed in N2 ways,⋯ , and a task Tn can be performed in Nn ways.
Then, the number of ways of performing the tasks T1 , T2 ,⋯ , Tn in succession is given
by the product: N1 N2 ⋯Nn .
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 2: A coin is tossed 3 times, and the sequence of heads and tails is recorded.
Determine the number of outcomes.
Example 3: An auto manufacturer has 3 different subcompact cars in the line.
Customers selecting one of these cars have a choice of 3 engine sizes, 4 body styles,
and 3 color schemes. How many different selections can a customer make?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 4: How many three-letter words that have exactly one vowel can be made
using the first seven letters of the alphabet where using a letter twice is permitted
but having two consonants next to each other is not?
Example 5: How many five-digit numbers can be formed if
a) Zero is not the first digit?
b) Zero cannot be the first digit and no digit can be repeated?
c) Zero cannot be the first digit, no digit can be repeated, and each number formed
must be even?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 6: Jack and Jill and 5 of their friends go to the movies. They all sit next
to each other in the same row. How many ways can this be done if
a) there are no restrictions?
b) Jill must sit in the middle?
c) Jill sits on one end of the row and Jack sits on the other end of the row?
d) Jack, Jill, or John sit in the middle seat?
e) Jack, Jill, and John sit in the middle seats?
f) Jack and Jill must sit next to each other?
g) Jill must not sit next to Jack?
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Notes for math 141
Section 6.1-6.3
Finite Mathematics
Example 7: Suppose we want to seat 12 people in a row of 12 seats. How many
arrangements are possible?
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