MATH 1420
INTRODUCTION TO COLLEGE MATH
CHAPTER 2 SUMMARY
Created 2/29/2012
Revised 5/29/2012
Section 2-1: The Basic Concepts of Set Theory
Set: a collection of objects
Element or member of a set: symbol  the objects belonging to a set
example: 7  { 5, 6, 7, 8 } note: 7 does not have braces
Three methods to designate a set:
1. Word description: the set of …
example: The set of even counting numbers less than 8.
2. Listing method: must have { } around each set, separate each element by a comma.
example: { 2, 4, 6 }
3. Set builder notation: { x | x is … }
example: { x | x is an even counting number less than 8 }
read as: “the set of all x’s such that x is an even counting number less than 8”
Naming a set: use a capital letter
example: A = { 1, 2, 3 }
Empty set or null set: symbol { } or 
a set containing no elements
Cardinal number or cardinality of a set: the number of elements in a set notation: n (A)
Finite set: the cardinal number is a whole number (the set ends)
example: if A = {1, 2, 3, 4 }, then n(A) = 4
Infinite set: cardinal number cannot be found (the set does not end)
example: B = { 1, 2 ,3, 4, … } Note that n(B) cannot be found.
Equal sets: the sets have the exact same elements (an element may be repeated)
examples: { 4, 5, 6 } = { 6, 4, 5 }
{ 4, 5, 6 } = { 4, 6, 6, 5, 4 }
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Section 2-2: Venn Diagrams and Subsets
Universal set: represented by a rectangle
symbol: uppercase U
Complement of a set: symbol A the set of elements that are in U but not in A.
The region outside set A, but inside the Universal set:
A
U
A
Use the following sets for the following definitions and examples:
U = { 1, 2, 3, 4, 5, 6, 7, 8 }
A = { 2, 4, 6 }
B = { 2, 3, 4, 5 }
C = { 2, 4 }
A = {1, 3, 5, 7, 8 }
B  = { 1, 6, 7, 8 }
C  = { 1, 3, 5, 6, 7, 8 }
Subset: symbol:  : A  B if every element of set A is a member of set B
examples: A  U, B  U, C  A
A  A since every set is a subset of itself
{4}  A, but 4  A since the “4” must have braces
Proper subset: symbol  : A  B if every element of set A is a member of set B, but A cannot equal B
examples: A  U, C  A
A  A since a set cannot be a proper subset of itself
Null set is a subset of every set (notation for null set: { } or  )
examples:   A,    ,   A
Number of subsets: If set A has n elements, then it has 2 n subsets.
Number of proper subsets: If set A has n elements then it has 2 n  1 proper subsets.
example: For set A, n(A) = 3, so set A has 2 3  8 subsets, and 2 3  1  7 proper subsets.
Listing subsets:
example: List all the subsets of set A:  , {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}
List all the proper subsets of set A:  , {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}
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Section 2-3: Set Operations and Cartesian Products
Use the following sets for the following definitions and examples:
A = {1, 2, 3, 4, 5, 6 }
B = {2, 3, 4, 7 }
Intersection of sets: written: A  B the set of elements common to both set A and B
example:
A  B = { 2, 3, 4 }
Union of sets: written A  B the set of elements that are in set A or set B or both
example:
A  B = { 1, 2, 3, 4, 5, 6, 7 }
Difference of sets: written A – B
examples: A – B = {1, 5, 6 }
B – A = {7}
Ordered pair: written: ( a, b ) where “a” is the first component and “b” is the second
note: ( a, b )  ( b, a )
Cartesian Product: written A  B
example: Let A = {1, 3, 5 } and B = { c, d }
then A  B = { (1, c), (1, d), (3, c), (3, d), (5, c), (5, d) } Note: must have ( ) around
the ordered pairs, and
must have { } around the set.
note: A  B  B  A
Cardinal Number of a Cartesian Product:
example: Let n(A) = 3 and n(B) = 7,
then n(A  B) = n(A)  n(B) = 3  7 = 21
or n(B  A) = n(B)  n(A) = 7  3 = 21
Venn Diagrams: To determine which region to shade in a Venn diagram, find the union or intersection
of the regions I, II, III, IV first, then shade in the appropriate region(s).
example: Shade A  B 
A consists of regions I, II, and B  consists of regions I, IV
The intersection is region I, so shade in region I.
U
A
I
B
II
III
IV
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Section 2-4: Surveys and Cardinal Numbers:
Cardinal Number Formula:
Example 1: Find n(A  B),
n(A  B) = n(A) + n(B) – n(A  B)
if n(A) = 15 n(B) = 7 and n(A  B) = 3
n(A  B) = n(A) + n(B) – n(A  B)
n(A  B) = 15 + 7 – 3
n(A  B) = 19
Using the formula:
or using the Venn Diagram:
The numbers in the regions of this diagram represent cardinalities.
U
A
12
3
B
To find n(A  B), add up the numbers in regions I, II, III
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12 + 3 + 4 = 19
Example 2: Find n(A), if n(A  B) = 35, n(A  B) = 12, n(B) =16
n(A  B) = n(A) + n(B) – n(A  B)
35 = n(A) + 16 – 12
35 = n(A) + 4
35 – 4 = n(A)
31 = n(A)
Using the formula:
or using the Venn Diagram:
The numbers in the regions of this diagram represent cardinalities.
U
A
B
12 4
First find the cardinality for region III: n(B) = 16, n(A  B) = 12,
Cardinality of region III = 16 – 12 = 4
Since n(A  B) = 35, cardinality of region I = 35 – 12 – 4 = 19
Therefore, n(A) = cardinality of regions I and II = 19 + 12 = 31
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