Chapter 1: Sets and Probability
Section 1.1-1.2
is a collection of items. Each item in a set is referred to as an
A
or a
.
Note:
• Sets are usually denoted by uppercase letters: A, B, C, ...
• Elements are usually denoted by lowercase letters: a, b, c, ...
• If a is an element of the set A, then we write a ∈ A.
• If a is NOT an element of the set A, then we write a ∈
/ A.
Curly brackets will indicate a set. When all the elements of a set are written out we refer
to this as
.
We can also define a set in terms of its properties. This is called
.
Example 1 Write the set containing the first five letters of the alphabet.
Roster Notation:
Set-builder Notation:
If every element of a set A is also an element of another set B, we say that A is a
of B, and we write
not a subset of B, we write
. If A is
.
1
Two sets A and B are
, written
if A ⊆ B and B ⊆ A (they have exactly the same elements).
If A ⊆ B but A 6= B, then we say A is a
of B.
Note: Be aware of the difference between ∈ and ⊆. We use ∈ when talking about an
ELEMENT being in a set. We use ⊆ when talking about a SET being a subset of another
set.
, written ∅ or {}, is the set with no elements.
The
Note: For any set, S, ∅ ⊆ S.
Example 2 Find all the subsets of A = {a, b, c}.
Note: There are 8 subsets and 7 proper subsets. In general, a set with n elements will
have
subsets and
proper subsets.
The set of all elements being considered is called the
We denote this set by U .
set.
A
is a way of visualizing sets. The universal
set is represented by a rectangle and sets are represented as circles inside the universal
set.
2
Example 3 This is a Venn diagram that represents A ⊆ B:
U
B
A
Set Operations
Given a universal set U and a set A ⊆ U , the
written
,
, is the set of all elements in U but not in A
U
A
Example 4 Let U = {a, b, c, e, g, i, k, m, o, r, s, u, w}, A = {a, e, i, o, u}, and B = {b, c, k, m, o}.
Find the following:
a) AC =
b) B C =
c) U C =
d) ∅C =
3
e) (AC )C =
Complement Rules: If U is a universal set and A is any set in the universal set, we
always have
(1) U C = ∅
(2) ∅C = U
(3) (AC )C = A
Note: The complement of a set is analogous to negation (NOT) in logic.
of two sets A and B, written
The
,
is the set of all elements that belong to A, or to B, or to both.
U
A
B
Example 5 Let U = {a, b, c, e, g, i, k, m, o, r, s, u, w}, A = {a, e, i, o, u}, and B = {b, c, k, m, o}.
Find the following:
a) A ∪ B =
b) A ∪ AC =
Note: In general, A ∪ AC = U . The union of sets is analogous to the disjunction (OR)
in logic.
The
of two sets A and B, written
4
,
is the set of all elements that belong to both the set A and the set B.
U
A
B
Example 6 Let A = {a, e, i, o, u}, B = {b, c, k, m, o}, and C = {d, k, m, r, t}. Find the
following:
a) A ∩ B =
b) A ∩ C =
Note: The intersection of sets is analogous to the conjunction (AND) in logic.
if they have no elements in common, that is
Two sets are
if
U
A
B
5
Laws for Set Operations:
(1) A ∪ B = B ∪ A (Union of sets is commutative)
(2) A ∩ B = B ∩ A (Intersection of sets is commutative)
(3) A ∪ (B ∪ C) = (A ∪ B) ∪ C (Union of sets is associative)
(4) A ∩ (B ∩ C) = (A ∩ B) ∩ C (Interesection of sets is associative)
(5) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (Unions distribute over intersections)
(6) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Intersections distribute over unions)
(7) (A ∪ B)C = AC ∩ B C (DeMorgan’s Law)
(8) (A ∩ B)C = AC ∪ B C (DeMorgan’s Law)
Example 7 Let U = {n, 2, 3, 4, w, 6, 7, 8, 9}, A = {n, w, 2, 8},
B = {x|x is an even number between 1 and 9}, C = {n, 3, 6, 8, 9}. Find
a) A ∩ (B ∪ C)C
b) (AC ∩ C) ∪ B C
c) (A ∪ B ∪ C)C
Example 8 Shade the appropriate region in the Venn Diagram
a) A ∩ B C ∩ C C
U
A
B
C
6
b) (A ∩ C) ∪ (B ∩ C C )
U
A
B
C
c) (B C ∪ C) ∩ (A ∪ C C )
U
A
B
C
d) (AC ∪ B)C ∩ C
U
A
B
C
The Number of Elements in a Set
The
elements in the set.
of a set A, denoted n(A), is the number of
7
U
A
n(A)
n(AC )
Example 9 The set A = {a, 2, 4, e} has
elements in it.
If B = {x|x is an even number between 1 and 9} has
elements in it.
Counting Rules:
(1) n(∅) = 0
(2) If A and B are disjoint, then
Example Let A = {a, b, c} and B = {d, e}. Find n(A), n(B), and n(A ∪ B).
U
A
B
(3) n(A) + n(AC ) = n(U )
(4) The Union Rule: For any sets A and B,
8
Example Let A = {a, b, c} and B = {b, d}. Find n(A), n(B), n(A ∪ B), and
n(A ∩ B).
U
A
B
(5) The Union Rule for 3 sets: For any sets A, B, and C,
U
A
B
C
Example 10 Given n(U ) = 75, n(A) = 40, n(B) = 45, and n(A ∪ B C ) = 45, find
n(A ∪ B).
9
Example 11 Given n(U ) = 100, n(A) = 40, n(B) = 37, n(C) = 35, n(A ∩ B) = 25,
n(A ∩ C) = 22, n(B ∩ C) = 24, and n(A ∩ B ∩ C C ) = 10, find n((A ∪ B ∪ C)C ).
Example 12 A recent survey of 200 children found that 150 liked brownies, 90 liked ice
cream, and 190 liked brownies or ice cream.
U
B
I
a) How many children liked brownies and ice cream?
b) How many children liked exactly one of these two desserts?
c) How many children liked brownies or did not like ice cream?
10
Example 13 A survey of 150 readers was conducted.
Let F be the set of readers who enjoy fantasy novels.
Let M be the set of readers who enjoy mystery novels.
Let C be the set of readers who enjoy classical literature.
Use the surevy results below to find the number of people in each section of the “given”
Venn diagram.
• 10 enjoy all three genres
• 36 enjoy at least two of these genres
• 68 enjoy mystery novels
• 5 enjoy fantasy and mystery, but not classical literature
• 14 do not enjoy any of these genres
• 13 enjoy fantasy and classical literature
• 98 enjoy mystery or classical literature, but not fantasy
U
F
M
C
a) Find the number of people who enjoy fantasy.
b) Find the number of people who enjoy at least one of these genres.
c) Find the number of people who enjoy fantasy or classical literature, but not mystery.
11