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Copyright © 2009 Pearson Education, Inc.
Chapter 1 Section 1 - Slide 1
Chapter 1.1
Chapter 2.1, 2.2
Critical Thinking Skills
Set Concepts and Subsets
Copyright © 2009 Pearson Education, Inc.
Chapter 1 Section 1 - Slide 2
WHAT YOU WILL LEARN
• Inductive and deductive reasoning
processes
• Methods to indicate sets, equal sets,
and equivalent sets
• Subsets and proper subsets
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Chapter 1 Section 1 - Slide 3
Section 1.1
Inductive Reasoning
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Chapter 1 Section 1 - Slide 4
Natural Numbers

The set of natural numbers is also called the set
of counting numbers.
N = {1,2,3,4,5,6,7,8,...}

The three dots, called an ellipsis, mean that 8 is
not the last number but that the numbers
continue in the same manner.
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Chapter 1 Section 1 - Slide 5
Divisibility
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

If a  b has a remainder of zero, then a is
divisible by b.
The even counting numbers are divisible by 2.
They are 2, 4, 6, 8,… .
The odd counting numbers are not divisible by
2. They are 1, 3, 5, 7,… .
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Chapter 1 Section 1 - Slide 6
Inductive Reasoning



The process of reasoning to a general
conclusion through observations of specific
cases.
Also called induction.
Often used by mathematicians and scientists to
predict answers to complicated problems.
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Chapter 1 Section 1 - Slide 7
Scientific Method


Inductive reasoning is a part of the scientific
method.
When we make a prediction based on specific
observations, it is called a conjecture.
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Chapter 1 Section 1 - Slide 8
Counterexample

In testing a conjecture, if a special case is found
that satisfies the conditions of the conjecture but
produces a different result, that case is called a
counterexample.
 Only one exception is necessary to prove a
conjecture false.
 If a counterexample cannot be found, the
conjecture is neither proven nor disproven.
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Chapter 1 Section 1 - Slide 9
Deductive Reasoning



A second type of reasoning process.
Also called deduction.
Deductive reasoning is the process of reasoning
to a specific conclusion from a general
statement.
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Chapter 1 Section 1 - Slide 10
Example: Inductive Reasoning


Use inductive reasoning to predict the next
three numbers in the pattern (or sequence).
7, 11, 15, 19, 23, 27, 31,…
Solution:
We can see that four is added to each term to
get the following term.
31 + 4 = 35,
35 + 4 = 39,
39 + 4 = 43
Therefore, the next three numbers in the
sequence are 35, 39, and 43.
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Chapter 1 Section 1 - Slide 11
Section 2.1
Set Concepts
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Chapter 1 Section 1 - Slide 12
Set

A collection of objects, which are called
elements or members of the set.

Listing the elements of a set inside a pair of
braces, { }, is called roster form.

The symbol , read “is an element of,” is used
to indicate membership in a set.

The symbol  means “is not an element of.”
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Chapter 1 Section 1 - Slide 13
Well-defined Set
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

A set which has no question about what
elements should be included.
Its elements can be clearly determined.
No opinion is associated its the members.
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Chapter 1 Section 1 - Slide 14
Roster Form

This is the form of the set where the elements
are all listed, separated by commas.
Example:
Set A is the set of all natural numbers less than
or equal to 25.
Solution: A = {1, 2, 3, 4, 5,…, 25}
The 25 after the ellipsis indicates that the
elements continue up to and including the
number 25.
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Chapter 1 Section 1 - Slide 15
Set-Builder (or Set-Generator) Notation
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
A formal statement that describes the members
of a set is written between the braces.
A variable may represent any one of the
members of the set.
Example: Write set B = {2, 4, 6, 8, 10} in setbuilder notation.
Solution:


B  x x N and x is an even number  10 .
The set of all x such that x is a natural number and x is an even number £ 10.
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Chapter 1 Section 1 - Slide 16
Finite Set
A set that contains no elements or the number
of elements in the set is a natural number.
Example:
Set S = {2, 3, 4, 5, 6, 7} is a finite set because
the number of elements in the set is 6, and 6 is
a natural number.

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Chapter 1 Section 1 - Slide 17
Infinite Set

An infinite set is a set where the number of
elements is not  or a natural number; that is,
you cannot count the number of elements.

The set of natural numbers is an example of an
infinite set because it continues to increase
forever without stopping, making it impossible to
count its members.
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Chapter 1 Section 1 - Slide 18
Equal Sets

Equal sets have the exact same elements in
them, regardless of their order.

Symbol:

Example: { 1, 5, 7 } = { 5, 7, 1 }
A=B
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Chapter 1 Section 1 - Slide 19
Cardinal Number

The number of elements in set A is its cardinal
number.

Symbol: n(A)

Example:
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A = { 1, 5, 7, 10 }
n(A) = 4
Chapter 1 Section 1 - Slide 20
Equivalent Sets

Equivalent sets have the same number of
elements in them.

Symbol: n(A) = n(B)

Example:
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A = { 1, 5, 7 } , B = { 2, 3, 4 }
n(A) = n(B) = 3
So A is equivalent to B.
Chapter 1 Section 1 - Slide 21
Empty (or Null) Set

The null set (or empty set ) contains absolutely
NO elements.

Symbol:
 or
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 
Chapter 1 Section 1 - Slide 22
Universal Set

The universal set contains all of the possible
elements which could be discussed in a
particular problem.

Symbol: U
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Chapter 1 Section 1 - Slide 23
Section 2.2
Subsets
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Chapter 1 Section 1 - Slide 24
Subsets


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A set is a subset of a given set if and only if all
elements of the subset are also elements of
the given set.
Symbol: 
To show that set A is not a subset of set B, one
must find at least one element of set A that is
not an element of set B. The symbol for “not a
subset of” is  .
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Chapter 1 Section 1 - Slide 25
Determining Subsets
Example:
Determine whether set A is a subset of set B.
A = { 3, 5, 6, 8 }
B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Solution:
All of the elements of set A are contained in set
B, so A  B.
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Chapter 1 Section 1 - Slide 26
Proper Subset

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All subsets are proper subsets except the
subset containing all of the given elements,
that is, the given set must contain one
element not in the subset (the two sets
cannot be equal).
Symbol: 
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Chapter 1 Section 1 - Slide 27
Determining Proper Subsets
Example:
Determine whether set A is a proper subset of
set B.
A = { dog, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B,
and sets A and B are not equal, therefore A  B.
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Chapter 1 Section 1 - Slide 28
Determining Proper Subsets (continued)
Example:
Determine whether set A is a proper subset of
set B.
A = { dog, bird, fish, cat }
B = { dog, cat, bird, fish }
Solution:
All the elements of set A are contained in set B,
but sets A and B are equal, therefore A  B.
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Chapter 1 Section 1 - Slide 29
Number of Distinct Subsets
The number of distinct subsets of a finite
set A is 2n, where n is the number of
elements in set A.
Example:

Determine the number of distinct subsets
for the given set { t , a , p , e }.

List all the distinct subsets for the given set:
{ t , a , p , e }.

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Chapter 1 Section 1 - Slide 30
Number of Distinct Subsets continued
Solution:

Since there are 4 elements in the given set,
the number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16 subsets.

{t,a,p,e},
{t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},
{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e},
{t}, {a}, {p}, {e}, { }
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Chapter 1 Section 1 - Slide 31