MM150
SURVEY OF MATHEMATICS
Unit 2 Seminar - Sets
SECTION 2.1: SET CONCEPTS

A set is a collection of objects.

The objects in a set are called elements.

Roster form lists the elements in brackets.
SECTION 2.1: SET CONCEPTS
Example: The set of months in the year is:
M = { January, February, March, April, May, June, July, August,
September, October, November, December }
Example: The set of natural numbers less than ten is:
SECTION 2.1: SET CONCEPTS

The symbol Є means “is an element of”.
Example: March Є
{ January, February, March, April }
Example: Kaplan Є { January, February, March, April }
SECTION 2.1: SET CONCEPTS

Set-builder notation doesn’t list the elements.
It tells us the rules (the conditions) for being in
the set.
Example: M = { x | x is a month of the year }
Example: A = { x | x Є N and x < 7 }
SECTION 2.1: SET CONCEPTS
Sample: A = { x | x Є N and x < 7 }
Example: Write the following using Set Builder Notation.
K = { 2, 4, 6, 8 }
SECTION 2.1: SET CONCEPTS
Sample : A = { x | x Є N and x < 7 }
Example: Write the following using Set Builder Notation.
S = { 3, 5, 7, 11, 13 }
SECTION 2.1: SET CONCEPTS

Set A is equal to set B if and only if set A and set
B contain exactly the same elements.
Example:
A = { Texas, Tennessee }
B = { Tennessee, Texas }
C = { South Carolina, South Dakota }
What sets are equal?
SECTION 2.1: SET CONCEPTS

The cardinal number of a set tells us how
many elements are in the set. This is denoted by
n(A).
Example:
What is n(A)?
n(B)?
n(C)?
A = { Ohio, Oklahoma, Oregon }
B = { Hawaii }
C = { 1, 2, 3, 4, 5, 6, 7, 8 }
SECTION 2.1: SET CONCEPTS

Set A is equivalent to set B if and only if n(A) =
n(B).
Example:
A = { 1, 2 }
B = { Tennessee, Texas }
C = { South Carolina, South Dakota }
D = { Utah }
What sets are equivalent?
SECTION 2.1: SET CONCEPTS

The set that contains no elements is called the
empty set or null set and is symbolized by { } or
Ø.
This is different from {0} and {Ø}!
SECTION 2.1: SET CONCEPTS

The universal set, U, contains all the elements
for a particular discussion.
We define U at the beginning of a discussion.
Those are the only elements that may be used.
SECTION 2.2: SUBSETS

Set A is a subset of set B, symbolized by A  B, if
and only if all the elements of set A are also in
set B.
orange
B =
yellow
red
purple
blue
green
SECTION 2.2: SUBSETS
B =
D=
Mom
Dad
Dad
Sister
Brother
Brother
SECTION 2.2: SUBSETS
7
3
B =
4
5
1
13
3
A =
1
C =
1
4
6
13
SECTION 2.2: SUBSETS
12
4
B =
8
6
2
10
4
A =
10
2
6
12
8
10
C =
6
8
SECTION 2.2: SUBSETS

Set A is a subset of set B, symbolized by A  B, if
and only if all the elements of set A are also in
set B.
Example:
Is A  B?
Is B  A?
A = { Vermont, Virginia }
B = { Rhode Island, Vermont, Virginia }
SECTION 2.2: SUBSETS

Set A is a proper subset of set B, symbolized by
A  B, if and only if all the elements of set A are
in set B and set A ≠ set B.
A =
1, 2, 3
B = 1, 2, 3, 4, 5
C = 1, 2, 3
SECTION 2.2: SUBSETS

Set A is a proper subset of set B, symbolized by
A  B, if and only if all the elements of set A are
in set B and set A ≠ set B.
Example:
Is A  B?
Is B  C?
A = { a, b, c }
B = { a, b, c, d, e, f }
C = { a, b, c, d, e, f }
SECTION 2.2: SUBSETS

The number of subsets of a particular set is
determined by 2n, where n is the number of
elements.
Example:
A = { a, b, c }
B = { a, b, c, d, e, f }
C={ }
How many subsets does A have?
B?
C?
SECTION 2.2: SUBSETS
Example: List the subsets of A.
A = { a, b, c }
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

A Venn diagram is a picture of our sets and
their relationships.
A
C
B
C
A
A
B
B
C
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

The complement of set A, symbolized by A′, is
the set of all the elements in the universal set
that are not in set A.
Example:
U = { m | m is a month of the year }
A = { Jan, Feb, Mar, Apr, May, July, Aug, Oct, Nov }
What is A´ ?
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

The complement of set A, symbolized by A′, is
the set of all the elements in the universal set
that are not in set A.
Example:
What is A´ ?
U = { 2, 4, 6, 8, 10, 12 }
A = { 2, 4, 6 }
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

The intersection of sets A and B, symbolized by
A ∩ B, is the set of elements containing all the
elements that are common to both set A and B.
Example:
What is A ∩ B?
B ∩ C?
C ∩ A?
A = { pepperoni, mushrooms, cheese }
B = { pepperoni, beef, bacon, ham }
C = { pepperoni, pineapple, ham, cheese }
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

The union of sets A and B, symbolized by A U B, is
the set of elements that are members of set A or set B
or both.
Example:
What is A U B?
B U C?
C U D?
A = { Jan, Mar, May, July, Aug, Oct, Dec }
B = { Apr, Jun, Sept, Nov }
C = { Feb }
D = { Jan, Aug, Dec }
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

Special Relationship:
n(A U B) = n(A) + n(B) - n(A ∩ B)
B = { Max, Buddy, Jake, Rocky, Bailey }
G = { Molly, Maggie, Daisy, Lucy, Bailey }
A
B
SECTION 2.3: VENN DIAGRAMS AND SET
OPERATIONS

The difference of two sets A and B, symbolized
by A – B, is the set of elements that belong to set
A but not to set B.
Example:
What is A - B?
A = { n | n Є N, n is odd }
B = { n | n Є N, n > 10 }
SECTION 2.4: VENN DIAGRAMS WITH THREE
SETS AND VERIFICATION OF EQUALITY OF SETS
Procedure for Constructing a Venn Diagram
with Three Sets: A, B, and C
1.
Determine the elements in A ∩ B ∩ C.
2.
Determine the elements in A ∩ B, B ∩ C, and A ∩ C
(not already listed in #1).
3.
Place all remaining elements in A, B, C as needed (not
already listed in #1 or #2).
4.
Place U elements not listed.
SECTION 2.4: VENN DIAGRAMS WITH THREE
SETS AND VERIFICATION OF EQUALITY OF SETS
Venn Diagram with Three Sets: A, B, and C
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10}
B = {1, 2, 3, 4, 5}
C = {2, 3, 5, 7, 8}
U
A
B
1.
A∩B∩C
2.
A ∩ B, B ∩ C, and A ∩ C
3.
A, B, C
4.
U
C
SECTION 2.4: VENN DIAGRAMS WITH THREE
SETS AND VERIFICATION OF EQUALITY OF SETS
De Morgan’s Laws
1.
(A ∩ B)´ = A´ U B´
2.
(A U B)´ = A´ ∩ B´
THANK YOU!

Read Your Text

Use the MML Graded Practice

Read the DB

Email: ttacker@kaplan.edu