Download Document

Section 1.1
Inductive
Reasoning
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Inductive and deductive reasoning
processes
1.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 2.1
Set Concepts
2.1-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn

Equality of sets

Application of sets

Infinite sets
2.1-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Set
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

2.1-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Description of Sets
Write a description of the set containing
the elements Monday, Tuesday,
Wednesday, Thursday, Friday, Saturday,
Sunday.
2.1-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
Non-Examples
(1, 2, 3,) and [1, 2, 3]
2.1-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Roster Form of Sets
Express the following in roster form.
a) Set A is the set of natural numbers
less than 6.
2.1-24
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-25
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-26
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: 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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Set-Builder Notation
to Roster Form


Write set A  x x N and 2  x  8
in roster form.
2.1-31
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-34
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Infinite Set
2.1-36

A set that is not finite is said to be
infinite.

The set of counting numbers is an
example of an infinite set.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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 }
2.1-38
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
2.1-41
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-42
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
2.1-43
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-45
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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:
2.1-46
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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}
2.1-47
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
One-to-one Correspondence
Other one-to-one correspondences
between sets S and C are possible.
2.1-48
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Null or Empty Set
The set that contains no elements is
called the empty set or null set and
is symbolized by
  or .
2.1-49
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-50
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-51
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.1-52
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 2.2
Subsets
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Subsets and proper subsets
2.2-54
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Subsets
2.2-55

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.”
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Subsets
2.2-56

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.
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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}
2.2-57
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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).
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Determining Proper Subsets
Example:
Determine whether set A is a proper
subset of set B.
A = { dog, cat }
B = { dog, cat, bird, fish }
2.2-60
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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 }
2.2-61
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.2-63
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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}.
2.2-64
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
2.2-67
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Number of Distinct Proper Subsets
Example:
Determine the number of distinct
proper subsets for the given set
{t, a, p, e}.
2.2-68
Copyright 2013, 2010, 2007, Pearson, Education, Inc.