Topic 1: The nature of
mathematics
Mathematics in the Modern World
By
Mrs. Cristina H. Price
Virtual Class Policies
 Synchronous sessions will be conducted using MS Teams.
 Virtual classroom will officially open at least 5 minutes before the scheduled class.
 Video camera may be turned off while the teacher is presenting the lesson.
However, the camera and microphone may be requested to be turned on during
interactive discussion.
 Class attendance will be downloaded using MS Teams.
 Submission of class homeworks and independent activities will be via MS Teams’
CLASS NOTEBOOK.
 Handouts and Assessments will be available in the Open LMS (OLMS).
Target Learning Outcomes:
1. Identify patterns in nature and regularities in the world.
2. Articulate the importance of mathematics in one's life.
3. Argue about the nature of mathematics, what it is now, how it is expressed,
represented, and used.
4. Express appreciation for mathematics as a human endeavor
Synchronous class sessions
May 17 (Tuesday)
May 19 (Thursday)
May 24 (Tuesday)– Asynchronous class (with independent activity)
May 26 (Thursday) – Asynchronous class (with independent activity)
May 31 (Tuesday)
June 2 (Thursday)
June 7 (Tuesday)
June 9 (Thursday)
Patterns and Numbers in
Nature and the World
Studies reveal that mathematics originated out of
necessity and out of interest. An alternative thesis also
considers religious rituals, agriculture, trade, surveying as
contributors in the development of mathematics.
The annual inundation of the Nile Valley forced the
Egyptians to develop some system for redetermining
land markings.
The Fibonacci sequence may be found in various objects
in nature.
Fibonacci Sequence in
Nature
Another simple example in which it is possible
to find the Fibonacci sequence in nature is
given by the number of petals of flowers. Most
have three (like lilies and irises), five (parnassia,
rose hips) or eight (cosmea), 13 (some daisies),
21 (chicory), 34, 55 or 89 (asteraceae).
Fibonacci sequences appear in biological
setting in two consecutive fibonacci numbers,
such as branching of trees,arrangement of
leaves on stem,the fruitlets of pineapple ,the
flowering of artichoke & arrangement of pine
cone & family tree of honeybees.
Importance of
Mathematics in Real Life
Mathematics makes
our life orderly and prevents
chaos. Certain qualities that are
nurtured by mathematics are
power of reasoning, creativity,
abstract or spatial thinking,
critical thinking, problemsolving ability and even
effective communication skills.
How the nature of mathematics is expressed
and represented
Mathematics relies on both logic and creativity, and it is pursued
both for a variety of practical purposes and for its intrinsic interest.
For some people, and not only professional mathematicians, the
essence of mathematics lies in its beauty and its intellectual
challenge.
As most commonly interpreted in education, mathematical
representations are visible or tangible productions – such as diagrams,
number lines, graphs, arrangements of concrete objects or
manipulatives, physical models, mathematical expressions, formulas
and equations, or depictions on the screen of a computer or
calculator – that encode, stand for, or embody mathematical ideas or
relationships.
Definition of a set
• A set is a collection of distinct and well defined
objects. The objects can be called elements or
members of the set.
• A set does not list an element more than once since an
element is member of a set or not.
• Capital letters are normally used to identify a set.
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Examples:
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Set of real numbers
Set of integers
Set of natural numbers
Set of rational
numbers
• Set of irrational
numbers
• Set of prime numbers
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Set of Real numbers
Real Numbers
Irrational Numbers
Rational Numbers
Integers
Negative integers
Rational numbers that
are not integers
Whole numbers
Zero
Natural or Counting numbers
Prime numbers
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Set of Complex
Numbers
Rational
numbers
Real
numbers
Whole Nos.
Prime
Nos.
Counting
Nos
Zero
Integers
Nonintegers
(Fractions)
Neg. of the
counting nos.
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Ways to identify a set
1. A written description
A is the set of calendar months beginning with the letter J.
2. List or Roster method
A = {January, June, July}
3. Set-Builder Notation
A = {x|x is a calendar month beginning with the letter J}
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Describe the given illustration:
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Other symbols:
The empty set or null set is the set having no elements.
Notations used is { } or Ø.
Example:
B is the set of 5-sided squares.
B={ }
The symbols and  are used to indicate if an element is in a
given set.
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SET THEORY SYMBOLS
∈ “is an element of ”
∉ “is not an element of ”
⊂ “is a proper subset of ”
⊆ “is a subset of ”
⊄ “is not a subset of ”
∅ the empty set; a set with no elements
∩ intersection
∪ union
• A or A’ “the complement of A”; all elements not in A
• n(A) “the number of elements in A”
• A = B “A is equal to B”; A and B contain the same elements
A  B A is equivalent to B”; A and B contain the same
number of elements
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Cardinality or cardinal number of a set
Notation used: n(A) or |A|.
A = {1, 2, 3, 5, 8, 9}
n(A)=|A| = 6
B = {2, 3, 8, 11, 12, 13}
n(B)=|B| = 6
C = {1, 2, 15}
n(C)=|C| = 3
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Equivalent sets
Two sets are equivalent if they contain the same
number of elements. If two sets are equivalent,
they can be placed in a one-to-one
correspondence. If two sets are equivalent, we
write
AB
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Equal Sets
Two sets are equal if they contain the exact same elements. The order
can be different.
A = {a, b, c, d}
B = {a, b, c}
C = {d, b, c, a}
A=C
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Finite and infinite sets
Finite sets – sets with certain number of elements.
Infinite sets – sets with unlimited or uncountable number
of elements.
The cardinality of the empty set is zero.
The cardinality of an infinite set is uncountable.
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Subsets
• If every member of set A is also a member of set B, then A is a
subset of B.
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A B
OR
BA
We can also say A is contained in B.
The alternative notation is:
It means B is a superset of A.
If A is not a subset of B, we write
A B
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Proper Subset
• If A is a subset of B ( A B), but A is not equal to B, then
we say A is a proper subset of B written
AB
• Examples:
A = {a, b, c, d, e}
B = {b, c, d}
C = {a, d, e, b, c}
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Examples:
A = {1, 2, 3, 5, 8, 9}
B = {2, 3, 8, 11, 12, 13}
C = {1, 2, 8}
AB
CA
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Other examples:
Use  and where appropriate.
A is the set of people living in Cubao, Quezon City, Philippines.
B is the set of people living in Quezon City, Philippines.
C is the set of weekdays.
D = {Monday, Tuesday, Wednesday, Thursday, Friday}
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More on subsets:
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The empty set is a subset of any set. Ø  S
The empty set is a proper subset of all sets except Ø.
n
2
If set A has n elements, it has
subsets.
If set A has n elements, it has 2 n 1 proper subsets.
The power set
 is a collection of
all the subsets of a given set.
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Determine the number of subsets and proper
subsets of the given sets.
1. A = {a, b}
2. M = {1, 2, 3}
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More on sets:
• Universal set U – the set that contains all the elements or
objects being considered.
• Complement of a set – contains the elements of
the
universal set that are not found in the given set
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Venn Diagrams and Set
Operations
• A Venn diagram is a visual diagram that shows the relationship of sets
with one another.
• The set of all elements being considered is called the universal set (U)
and is represented by a rectangle.
A’
A
U
• The complement of A, A’ , is the set of all elements of U but not in A.
• A’ = {x | x  U and x A}
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Disjoint sets
Set A and B are disjoint because they do not
share any common elements.
B
A
U
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Proper Subsets
B is a proper subset of A.
B
A
U
B  A and B  A  B  A
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Equal Sets
B is a subset of A and B = A.
A=B
U
If A  B, then A  B and B  A.
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Union of A and B
A B
The union of A and B is the set
of elements in set A or set B.
A  B  {x | x A or x  B}
B
A
U
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Intersection of A and B
The intersection of A and B is the set of all
elements in both set A and B.
A  B  {x | x  A and x  B}
B
A
U
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Examples
A = {Christian, Lois}
B = {Liza, Dawn}
C = {Marty, Liza, Lois}
AB ={Christian, Lois, Liza, Dawn}
A B = {
} or Ø
AC = {Lois}
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Subtraction
A – B is the set that contains all elements of A that are not in B.
A = {Mary, Mark, Fred, Angela, Frank, Laura}
B = {Fred, Mary, Frank, Jane}
A – B = {Mark, Angela, Laura}
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Venn Diagram
Problem 1:
150 college freshmen were interviewed.
• 85 were registered for a math class.
• 70 were registered for an English class.
• 50 were registered for both math and
M
E
English.
U
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Questions:
1.
2.
3.
4.
How many signed up only for a math class?
How many signed up only for an English class?
How many signed up for math or English?
How many signed up for neither math or English?
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Problem 2
200 individuals were interviewed.
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48 chose Red
52 chose Pink
65 chose Blue
15 chose Red and Pink
20 chose Red and Blue
12 chose Pink and Blue
10 chose all colors
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Questions:
1. How many individuals chose none of the
three colors?
2. How many individuals chose Red, but not
Pink or Blue?
3. How many individuals chose Pink and Red
but not Blue?
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Other problem using Venn diagram
Create a Venn diagram to show the relationship among the sets.
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U is the set of whole numbers from 0 to 15.
A is the set of multiples of 3.
B is the set of primes.
C is the set of odd numbers.
(Assume that A, B, & C are inside U)
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Appreciation of
Mathematics as a
Human Endeavor
Mathematics provides an effective
way of building
mental discipline and encourages
logical reasoning and mental rigor. In
addition, mathematical knowledge
plays a crucial role in understanding
the contents of other school subjects
such as science, social studies, and
even music and art.
Additional Resources for the Topic
• Read the book ‘The Number Devil’ by
Enzensberger
• Read the book ‘A Day’s Adventure in Math
Wonderland by Akiyama and Ruiz
• https://vimeo.com/9953368
End of Lesson Presentation
Caritas Christi Urget Nos!