Transcript 2.2 Day 1 Notes

Math in Our World
Section 2.2 D1
Subsets and Set Operations
Learning Objectives
Define the complement of a set.
Find all subsets of a set.
Use subset notation.
Find the number of subsets for a set.
Universal Set
A universal set, symbolized by U, is the set
of all potential elements under consideration
for a specific situation.
Once we define a universal set in a given setting, we
are restricted to considering only elements from that
set. If U = {1, 2, 3, 4, 5, 6, 7, 8}, then the only elements
we can use to define other sets in this setting are the
integers from 1 to 8.
Complement
The complement of a set A, symbolized A,
is the set of elements contained in the
universal set that are not in A.
U
A
A
This Venn Diagram shows the visual
representation of the sets U and A.
The complement of a set A is all the
things inside the rectangle, U, that
are not inside the circle representing
set A.
EXAMPLE 1
Finding the
Complement of a Set
Let U = {v, w, x, y, z} and A = {w, y, z}. Find A and
draw a Venn diagram that illustrates these sets.
SOLUTION
Using the list of elements
in U, we just have to cross
out the ones that are also
in A. The elements left
over are in A.
U = {v, w, x, y, z}
A = {v, x}
U
A
w z
y
v
x
Subsets
If every element of a set A is also an
element of a set B, then A is called a subset
of B. The symbol  is used to designate a
subset; in this case, we write A  B.
•Every set is a subset of itself. Every element of a set
A is of course an element of set A, so A  A.
•The empty set is a subset of every set.
EXAMPLE 2
Finding All Subsets of a Set
Find all subsets of A = {American Idol, Survivor}.
SOLUTION
Number of elements in Subset Subsets with that Number of Elements
2
1
{American Idol, Survivor}
{American Idol}, {Survivor}
0

So the subsets are: {American Idol, Survivor},
{American Idol}, {Survivor}, 
Proper Subsets
If a set A is a subset of a set B and is not
equal to B, then we call A a proper subset
of B, and write A  B.
The Venn diagram for a proper subset is shown below.
In this case, U = {1, 2, 3, 4, 5}, A = {1, 3, 5}, and
B = {1, 3}.
A
U
B
1 3
4
5
2
EXAMPLE 3
Finding Proper Subsets
of a Set
Find all proper subsets of {x, y, z}.
SOLUTION
Number of elements in Subset Subsets with that Number of Elements
3
2
1
{x, y, z}
{x, y}, {x, z}, {y, z}
{x}, {y}, {z}
0

We’ll eliminate
this one since
it’s equal to the
original.
So the proper subsets are: {x, y}, {x, z}, {y, z}, {x}, {y}, {z}, 
EXAMPLE 4
Understanding Subset Notation
State whether each statement is true or false.
(a){1, 3, 5}  {1, 3, 5, 7}
(b) {a, b}  {a, b}
(c) {x | x  N and x > 10}  N
(d) {2, 10}  {2, 4, 6, 8, 10} . - “not a subset of”
(e) {r, s, t}  {t, s, r}  - “not a subset of”
(f ) {Lake Erie, Lake Huron}  The set of Great
Lakes
EXAMPLE 4
Understanding Subset Notation
SOLUTION
(a) All of 1, 3, and 5 are in the second set, so {1, 3, 5} is a subset of
{1, 3, 5, 7}. The statement is true.
(b) Even though {a, b} is a subset of {a, b}, it is not a proper subset,
so the statement is false.
(c) Every element in the first set is a natural number, but not all
natural numbers are in the set, so that set is a proper subset of
the natural numbers. The statement is true.
(d) Both 2 and 10 are elements of the second set, so {2, 10} is a
subset, and the statement is false.
(e) The two sets are identical, so {r, s, t} is not a proper subset of {t,
s, r}. The statement is true.
(f ) Lake Erie and Lake Huron are both Great Lakes, so the
statement is true.
EXAMPLE 5
Understanding Subset Notation
State whether each statement is true or false.
(a)  {5, 10, 15}
(b) {u, v, w, x}  {x, w, u}
(c) {0}  
(d)   
EXAMPLE 5
Understanding Subset Notation
SOLUTION
(a) True: the empty set is a proper subset of every set.
(b) False: v is an element of {u, v, w, x} but not {x, w, u}.
(c) The set on the left has one element, 0. The empty
set has no elements, so the statement is false.
(d) The empty set is a subset of itself (as well as every
other set), but not a proper subset of itself since it is
equal to itself. The statement is false.
Number of Subsets for a Finite Set
If a finite set has n elements, then the set
has 2n subsets and 2n – 1 proper subsets.
Number of elements : n
Number of subsets : 2n
0
1
1
2
2
4
3
8
Number of proper subsets : 2n – 1
0
1
3
7
EXAMPLE 6
Finding the Number of
Subsets of a Set
Find the number of subsets and proper subsets of
the set {1, 3, 5, 7, 9, 11}.
SOLUTION
The set has n = 6 elements, so there are 2n, or 26 = 64,
subsets.
Of these, 2n – 1 , or 64 – 1 = 63, are proper.
Classwork
p. 63: 2, 11, 13, 15, 16, 18, 21, 23, 24, 25, 29, 30, 31,
32, 34, 35, 39, 40, 43