MATH 166 Spring 2016
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Chapter 1 Sets and Probability
1.1 Introduction to Sets
The Language of Sets
Definition: A set is a collection of items. These items are referred to as the elements or members
of the set. The empty set, written as ∅ or {}, is the set with no elements. Note that 0 6= ∅.
Example: The set containing the numbers 1,2, and 3 would be written {1, 2, 3}. Notice that the
set is contained in curly brackets.
Definition: When all the elements of the set are written out, we refer to this as roster notation. If
we wanted to refer to this set without writing all the elements, we could define the set in terms of
its properties. This is called set-builder notation.
Examples:
1. The set containing the first 10 letters in the English alphabet would be written as
{a, b, c, d, e, f, g, h, i, j}
in roster notation. We write
{x|x is one of the first 10 letters in the English alphabet}
in set-builder notation. This is read “the set of all x such that x is one of the first 10 letters in
the English alphabet”.
2. Let A = {a, e, l, t, r}. Which of the following sets are equal to A? (Select all that apply.)
(a) {x|x is a letter of the word late}
(b) {x|x is a letter of the word rated}
(c) {x|x is a letter of the word relate}
(d) {x|x is a letter of the word later}
(e) {x|x is a letter of the word latter}
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If we will be using a set more than once in a discussion, it is useful to define the set with a symbol,
usually an uppercase letter. So
S = {a, b, c, d, e, f, g, h, i, j}
We can say c is an element of the set S or simply write
c ∈ S.
The symbol ∈ is read “is an element of”. We can also say that the set R = {c} is a subset of our
larger set S as every element in the set R is also in the set S.
Definition: If every element of a set A is also an element of another set B, we say that A is a
subset of B and write A ⊆ B. If A is not a subset of B, we write A * B. If every element in the
set A is also in the set B but A 6= B, we say that A is a proper subset of B. This is written as A ⊂ B.
Remark: By the definition of subset, given any set A, we must have A ⊆ A and ∅ ⊆ A.
Example: If A = {1, 2, 3, 4}, determine whether the following statements are true or false.
1. 1, 3 ∈ A
2. {1, 3} ∈ A
3. {1, 5} ∈
/A
4. {1, 3} ⊆ A
5. 1, 3 ⊆ A
6. {1, 5} ⊆ A
7. {1, 2, 3, 4} ⊆ A
8. ∅ ∈ A
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MATH 166 Spring 2016
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Definition: The universal set is the set of all elements being considered and is denoted by U .
Example: If we are to take a national survey of voter satisfaction with the president, the universal
set is the set of all voters in this country.
A Venn diagram is a way of visualizing sets. The universal set is represented by a rectangle and
sets are represented as circles inside the universal set.
Example: The following figure is a Venn diagram that visualizes the concept that B ⊆ U and
A ⊆ B. The U above the rectangle will be dropped in later diagrams as we will abide by the
convention that the rectangle always represents the universal set.
Set Operations
There are three set operations: complement, union, and intersection.
Definition: Given a universal set U and a set A, the complement of A, written AC , is the set of all
elements that are in U but not in A, that is,
AC = {x|x ∈ U, x ∈
/ A}
Complement Rules: If U is a universe set, and A is any subset of a universal set U , we must
always have
U C = ∅, ∅C = U, (AC )C = A
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Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5}. Find AC , B C , U C , ∅C ,
and (AC )C in roster notation.
Definition: The union of two sets A and B, written A ∪ B, is the set of all elements that belong to
A, or to B, or to both. Thus
A ∪ B = {x|x ∈ A or x ∈ B or both}
= {x|x ∈ A or x ∈ B}
Remark: If U is a universal set and A ⊂ U , then A ∪ AC = U .
Example: Let U, A, B be defined as in previous example. Find A ∪ B and A ∪ AC .
Definition: The intersection of two sets A and B, written 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}
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Definition: Two sets A and B are disjoint if they have no elements in common, that is, if A∩B = ∅.
Remark: Since A and AC are disjoint, A ∩ AC = ∅.
Examples:
1. Let the universal set U = {a, b, c, d} with sets A = {a, b} and B = {b, c}. Find the set AC ∩ B.
Which of the sets below are disjoint to this set? (Select all that apply.)
(a) A ∩ B
(b) AC ∪ B
2. Let U be the set of all residents of your state and let A and H be defined as follows. Describe
the set (A ∪ H)C in words.
A = {x|x owns an automobile}
H = {x|x owns a house}
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3. Let U = {−10, −5, −2, 0, 5, 6, 10, 12, 17, 18}, A = {−10, −2, 5, 10, 17}, B = {−5, 0, 6, 12, 18},
and C = {−10, −5, 0, 5, 12, 17}. List the elements of each set.
(a) (A ∪ B)C ∩ C C
(b) AC ∩ (B ∩ C C )
Laws for Set Operations:
1. A ∪ B = B ∪ A
Commutative law for union
2. A ∩ B = B ∩ A
Commutative law for intersection
3. A ∪ (B ∪ C) = (A ∪ B) ∪ C
Associative law for union
4. A ∩ (B ∩ C) = (A ∩ B) ∩ C
Associative law for intersection
5. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributive law for union
6. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Distributive law for intersection
7. (A ∪ B)C = AC ∩ B C
De Morgan law
8. (A ∩ B)C = AC ∪ B C
De Morgan law
Proof:
1. (A ∪ B)C = AC ∩ B C
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2. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Examples:
1. Indicate where the given set AC ∩ B ∩ C is located on the figure below by clicking the boxes
within regions that should be shaded.
2. Let U be the set of all major league baseball players and let N, S, F , and H be defined as
follows.
N = {x|x plays for the New York Yankees}
S = {x|x plays for the San Francisco Giants}
F = {x|x is an outfielder}
H = {x|x has hit 20 homers in one season}
Write the set that represents the following description: San Francisco Giants who are outfielders
or who hit 20 homers in one season.
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