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Section 1.1
Inductive
Reasoning
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What You Will Learn
Inductive and deductive reasoning
processes
1.1-2
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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.
1.1-3
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Divisibility
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,… .
1.1-4
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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.
1.1-5
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Example 3: Inductive Reasoning
What reasoning process has led to the
conclusion that no two people have the
same fingerprints or DNA? This
conclusion has resulted in the use of
fingerprints and DNA in courts of law
as evidence to convict persons of
crimes.
1.1-6
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Example 3: Inductive Reasoning
Solution:
In millions of tests, no two people have
been found to have the same
fingerprints or DNA. By induction, then,
we believe that fingerprints and DNA
provide a unique identification and can
therefore be used in a court of law as
evidence. Is it possible that sometime
in the future two people will be found
who do have exactly the same
fingerprints or DNA?
1.1-7
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Scientific Method
Inductive reasoning is a part of the
scientific method.
When we make a prediction based on
specific observations, it is called a
hypothesis or conjecture.
1.1-8
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Example 5: Pick a Number, Any
Number
Pick any number, multiply the number
by 4, add 2 to the product, divide the
sum by 2, and subtract 1 from the
quotient. Repeat this procedure for
several different numbers and then
make a conjecture about the
relationship between the original
number and the final number.
1.1-9
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Example 5: Pick a Number, Any
Number
Solution:
Pick a number:
say, 5
Multiply the number by 4: 4 × 5 = 20
Add 2 to the product: 20 + 2 = 22
Divide the sum by 2: 20 ÷ 2 = 11
Subtract 1 from quotient: 11 – 1 = 10
1.1-10
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Example 5: Pick a Number, Any
Number
Solution:
We started with the number 5 and
finished with the number 10.
Start with the 2, you will end with 4.
Start with 3, final result is 6.
4 would result in 8, and so on.
We may conjecture that when you
follow the given procedure, the number
you end with will always be twice the
original number.
1.1-11
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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.
•
•
1.1-12
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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Deductive Reasoning
A second type of reasoning process is
called deductive reasoning.
Also called deduction.
Deductive reasoning is the process of
reasoning to a specific conclusion from
a general statement.
1.1-13
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Example 6: Pick a Number, n
Prove, using deductive reasoning, that
the procedure in Example 5 will always
result in twice the original number
selected.
Note that for any number n selected,
the result is 2n, or twice the original
number selected.
1.1-14
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Example 6: Pick a Number, n
Solution:
To use deductive reasoning, we begin with
the general case rather than specific
examples.
Pick a number: n
Multiply the number by 4: 4n
Add 2 to the product:
4n + 2
Divide the sum by 2: (4n + 2)÷2 = 2n + 1
Subtract 1 from quotient: 2n + 1 – 1 = 2n
1.1-15
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Section 2.1
Set Concepts
2.1-16
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What You Will Learn
Equality of sets
Application of sets
Infinite sets
2.1-17
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Set
•
•
•
•
•
2.1-18
A set is a collection of objects, which
are called elements or members of the
set.
Three methods of indicating a set:
Description
Roster form
Set-builder notation
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Well-defined Set
A set is well defined if its contents can
be clearly defined.
Example:
The set of U.S. presidents is a well
defined set. Its contents, the
presidents, can be named.
2.1-19
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Example 1: Description of Sets
Write a description of the set containing
the elements Monday, Tuesday,
Wednesday, Thursday, Friday, Saturday,
Sunday.
2.1-20
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Example 1: Description of Sets
Solution
The set is the days of the week.
2.1-21
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Roster Form
Listing the elements of a set inside a pair
of braces, { }, is called roster form.
Example
{1, 2, 3,} is the notation for the set
whose elements are 1, 2, and 3.
(1, 2, 3,) and [1, 2, 3] are not sets.
2.1-22
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Naming of Sets
Sets are generally named with capital
letters.
Definition: Natural Numbers
The set of natural numbers or counting
numbers is N.
N = {1, 2, 3, 4, 5, …}
2.1-23
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Example 2: Roster Form of Sets
Express the following in roster form.
a) Set A is the set of natural
numbers less than 6.
Solution:
a) A = {1, 2, 3, 4, 5}
2.1-24
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Example 2: Roster Form of Sets
Express the following in roster form.
b) Set B is the set of natural numbers
less than or equal to 80.
Solution:
b) B = {1, 2, 3, 4, …, 80}
2.1-25
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Example 2: Roster Form of Sets
Express the following in roster form.
c) Set P is the set of planets in Earth’s
solar system.
Solution:
c) P = {Mercury, Venus, Earth, Mars,
Jupiter, Saturn, Uranus, Neptune}
2.1-26
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Set Symbols
•
The symbol ∈, read “is an element of,”
is used to indicate membership in a
set.
•
The symbol ∉ means “is not an
element of.”
2.1-27
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Set-Builder Notation
(or Set-Generator Notation)
2.1-28
•
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.
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Example 4: Using Set-Builder
Notation
a) Write set B = {1, 2, 3, 4, 5} in
set-builder notation.
b) Write in words, how you would
read set B in set-builder notation.
2.1-29
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Example 4: Using Set-Builder
Notation
Solution
a)


or
B  x x N and x  5
B  x x N and x  6
b) The set of all x such that x is a
natural number and x is less than 6.
2.1-30
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Example 6: Set-Builder Notation
to Roster Form


Write set A  x x N and 2  x  8
in roster form.
Solution
A = {2, 3, 4, 5, 6, 7}
2.1-31
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Finite Set
A set that contains no elements or the
number of elements in the set is a
natural number.
Example:
Set B = {2, 4, 6, 8, 10} is a finite set
because the number of elements in the
set is 5, and 5 is a natural number.
2.1-32
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Infinite Set
A set that is not finite is said to be
infinite.
• The set of counting numbers is an
example of an infinite set.
•
2.1-33
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Equal Sets
Set A is equal to set B, symbolized
by A = B, if and only if set A and set
B contain exactly the same
members.
Example: { 1, 2, 3 } = { 3, 1, 2 }
2.1-34
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Cardinal Number
The cardinal number of set A,
symbolized n(A), is the number of
elements in set A.
Example:
A = { 1, 2, 3 } and
B = {England, Brazil, Japan}
have cardinal number 3,
n(A) = 3 and n(B) = 3
2.1-35
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Equivalent Sets
Set A is equivalent to set B if and only
if n(A) = n(B).
Example:
D={ a, b, c }; E={apple, orange, pear}
n(D) = n(E) = 3
So set A is equivalent to set B.
2.1-36
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Equivalent Sets - Equal Sets
Any sets that are equal must also be
equivalent.
• Not all sets that are equivalent are
equal.
Example:
D ={ a, b, c }; E ={apple, orange, pear}
n(D) = n(E) = 3; so set A is equivalent
to set B, but the sets are NOT equal
•
2.1-37
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One-to-one Correspondence
Set A and set B can be placed in oneto-one correspondence if every
element of set A can be matched with
exactly one element of set B and every
element of set B can be matched with
exactly one element of set A.
2.1-38
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One-to-one Correspondence
Consider set S states, and set C, state
capitals.
S = {North Carolina, Georgia, South
Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee,
Atlanta}
Two different one-to-one
correspondences for sets S and C are:
2.1-39
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One-to-one Correspondence
S = {No Carolina, Georgia, So Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee, Atlanta}
S = {No Carolina, Georgia, So Carolina, Florida}
C = {Columbia, Raleigh, Tallahassee, Atlanta}
2.1-40
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One-to-one Correspondence
Other one-to-one correspondences
between sets S and C are possible.
Do you know which capital goes with
which state?
2.1-41
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Null or Empty Set
The set that contains no elements is
called the empty set or null set and
is symbolized by
  or .
2.1-42
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Null or Empty Set
Note that {∅} is not the empty set.
This set contains the element ∅ and
has a cardinality of 1.
• The set {0} is also not the empty set
because it contains the element 0. It
has a cardinality of 1.
•
2.1-43
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Universal Set
The universal set, symbolized by U,
contains all of the elements for any
specific discussion.
• When the universal set is given, only
the elements in the universal set may
be considered when working with the
problem.
•
2.1-44
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Universal Set
Example
If the universal set is defined as
U = {1, 2, 3, 4, ,…,10}, then only the
natural numbers 1 through 10 may be
used in that problem.
2.1-45
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Section 2.2
Subsets
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What You Will Learn
Subsets and proper subsets
2.2-47
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Subsets
Set A is a subset of set B,
symbolized A ⊆ B, if and only if all
elements of set A are also elements
of set B.
• The symbol A ⊆ B indicates that
“set A is a subset of set B.”
•
2.2-48
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Subsets
The symbol A ⊈ B set A is not a
subset of set B.
• 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.
•
2.2-49
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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.
2.2-50
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Proper Subset
Set A is a proper subset of set B,
symbolized A ⊂ B, if and only if all
of the elements of set A are
elements of set B and set A ≠ B
(that is, set B must contain at least
one element not is set A).
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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.
2.2-52
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Determining Proper Subsets
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.
2.2-53
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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.
2.2-54
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Number of Distinct Subsets
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}.
2.2-55
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Number of Distinct Subsets
Solution:
Since there are 4 elements in the given
set, the number of distinct subsets is
24 = 2 • 2 • 2 • 2 = 16.
{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}, { }
2.2-56
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Number of Distinct Proper Subsets
The number of distinct proper
subsets of a finite set A is 2n – 1,
where n is the number of elements
in set A.
2.2-57
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Number of Distinct Proper Subsets
Example:
Determine the number of distinct
proper subsets for the given set
{t, a, p, e}.
2.2-58
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Number of Distinct Subsets
Solution:
The number of distinct proper subsets is
24 – 1= 2 • 2 • 2 • 2 – 1 = 15.
They are {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}, {
}.
Only {t,a,p,e}, is not a proper subset.
2.2-59
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