The Nature of Mathematics Mathematics helps usto take the complex processes that is naturally occuring growth of trees.

mplex processes that is naturally occuring growth of trees. There is also a flow pattern present in meandering rivers with
in the world around us and it represents them by utilizing logic to make things the repetition of undulating lines.
In the book of Stewart, Nature’s Number, he that mathematics is a formal more organized and more efficient. Further, mathematics also facilitate not
system of thought that was gradually developed in the human mind and only to weather, but also to control the weather ---- be it social, natural, Patterns of Movement. In the human walk, the feet strike the ground in a
evolved in the statistical, political, or medical. regular rhythm: the left-right-left-right-left rhythm. When a horse, a four-
human culture. legged creature walks, there is more of a complex but equally rhythmic
pattern. This prevalence of pattern in locomotion extends to the scuttling of
Mathematics is present in everything we do; it is all around us and it is the The Mathematical Landscape There is a region in the human mind that is insects, the flights of birds, the pulsations of jellyfish, and also the wave-like
building block of our daily activities. It has been at the forefront of each and capable of constructing and discerning the deepest insights being perceived movements of fish, worms, and snakes.
every period of our development, and as our civilized societies advanced, our from the natural world. In this region, the mathematical landscape exists-
needs of mathematics pioneering arose on the frontier of our course as we wherein concepts of numbers, symbols, equations, operations calculations, Patterns of Rhythm. Rhythm is conceivably the most basic pattern in nature.
prepare our human species to traverse the cosmic shore. abstractions, Our hearts and lungs follow a regular repeated pattern of sounds or
and proofs are the inhabitants as well as the constructs of the impenetrable movement whose timing is adapted to our body’s needs. Many of nature’s
Mathematics is a Tool vastness of its unchartered territories. In this landscape, a number is not simply rhythms are most likely similar to a heartbeat, while others are like breathing.
Mathematics, as a tool, is immensely useful, practical, and powerful. It is not a mathematical tree of counting. Also, infinite variables can be encapsulate to The beating of the heart, as well as breathing, have a default pattern.
about crunching numbers, formulas, and symbols but rather, it is all about finite.
forming new ways to see problems so we can understand them by combining Patterns of Texture. A texture is a quality of a certain object that we sense
insights with imagination. It also allows us to perceive realities in different And beneath the surface of this mathematical landscape are firmly-woven through touch. It exists as a literal surface that we can feel, see, and imagine.
contexts that would otherwise be intangible to us. It can be likened to our proofs, theorems, definitions, and axioms which are intricately “fertilized” by Textures are of many kinds. It can be bristly, and rough, but it can also be
sense of sight and touch. Mathematics is our sense to decipher patterns, reasoning, analytical, critical thinking and germicide by mathematical logic that smooth, cold, and hard.
relationships, and logical connections. It is our whole new way to see and made them precise, exact and powerful.
understand the modern world. Geometric Patterns. A geometric pattern is a kind of pattern which consists of
How Mathematics is Done a series of shapes that are typically repeated. These are regularities in the
It is done with curiosity, with a penchant for seeking patterns and generalities, natural world that are repeated in a predictable manner. Geometrical patterns
with a desire to know the truth, with trial and error, and without fear of facing are usually visible on cacti and succulents
more questions and problems to solve.
Waves and Dunes
Mathematics is for Everyone A wave is any form of disturbance that carries energy as it moves. Waves are of
Everyone uses mathematics, whoever they are, wherever they are, and different kinds: mechanical waves which propagate through a medium ---- air
whenever they need to. From mathematicians to scientists, from professionals or
to ordinary people, they all water, making it oscillate as waves pass by.Wind waves, on the other hand, are
use mathematics. For mathematics puts order amidst disorder. It helps us surface waves that create the chaotic patterns of the sea. Similarly, water
become better persons and helps make the world a better place to live in. waves are created by energy passing through water causing it to move in a
circular motion. Likewise, ripple patterns and dunes are formed by sand wind
Mathematics is a set of problem-solving tools. It provides answers to existing
as they pass over the sand
questions and presents solutions to occurring problems. It has the power to
PATTERN
unveil the reasons behind occurrences and it offers explanations. Moreover,
mathematics, as a study of patterns, allows people to observe, hypothesize,
The mathematics in our world is rooted in patterns. Patterns are all around us. Spots and Stripes
experiment, discover, and recreate. On the other hand, mathematics is an art
Finding and understanding patterns give us great power to play like god. With We can see patterns like spots on the skin of a giraffe. On the other hand,
and a process of thinking. For it involves reasoning, which can be inductive or
patterns, we can discover and understand new things; we learn to predict and stripes are visible on the skin of a zebra. Patterns like spots and stripes that are
deductive, and it applies methods of proof both in fashion that is conventional
ultimately control the future for our own advantage. A pattern is a structure, commonly present in different organisms are results of a reaction-diffusion
and unventional.
form, or design that is regular, consistent, or recurring. Patterns can be found system (Turing, 1952). The size and the shape of the pattern depend on how
in nature, in human-made designs, or in abstract ideas. fast the chemicals diffuse and how strongly they interact.
Mathematics is Everywhere
Spirals
We use mathematics in their daily tasks and activities. It is our important tool
Different Kinds of Pattern The spiral patterns exist on the scale of the cosmos to the minuscule forms of
in the field of sciences, humanities, literature, medicine, and even in music and
Patterns of Visuals. Visual patterns are often unpredictable, never quite microscopic animals on earth. The Milky Way that contains our Solar System is
arts; it is in the rhythm of our daily activities, operational in our communities,
repeatable, and often contain fractals. These patterns are can be seen from the a barred spiral galaxy with a band of bright stars emerging from the center
and a default system of our culture. There is mathematics wherever we go. It
seeds and pinecones to the branches and leaves. They are also visible in self- running across the middle of it. Spiral patterns are also common and
helps us cook delicious meals by exacting our ability to measure and
similar noticeable among plants and some animals. Spirals appear in many plants such
moderately control of heat. It also helps us to shop wisely, read maps, use the
replication of trees, ferns, and plants throughout nature. as pinecones, pineapples, and sunflowers. On the other hand, animals like ram
computer, remodel a home with constrained budget with utmost economy.
and kudu also have spiral patterns on their horns.
Patterns of Flow. The flow of liquids provides an inexhaustible supply of
nature’s patterns. Patterns of flow are usually found in the water, stone, and
The Essential Roles of Mathematics
even in the Symmetries
In mathematics, if a figure can be folded or divided into two with two halves three dots at the end of the visible patterns means that the sequence is • Enables both the teacher and the students to communicate mathematical
which are the same, such figure is called a symmetric figure infinite. knowledge with precision

Reflection symmetry, sometimes called line symmetry or mirror symmetry, Arithmetic sequence. It is a sequence of numbers that follows a definite C. Comparison of Natural Language into Mathematical Language
captures symmetries when the left half of a pattern is the same as the right pattern. To determine if the series of numbers follow an arithmetic sequence, The table below is an illustration on the comparison of a natural language
half. check the difference between two consecutive terms. If common difference is (expression or sentence) to a mathematical language.
Rotations, also known as rotational symmetry, captures symmetries when it observed, then definitely arithmetic sequence governed the pattern
still looks the same after some rotation (of less than one full turn). T
Translations. This is another type of symmetry. Translational symmetry exists Geometric sequence. If in the arithmetic sequence we need to check for the D. Expressions versus Sentences
in patterns that we see in nature and in man-made objects. Translations common difference, in geometric sequence we need to look for the common Ideas regarding sentences:
acquire symmetries when units are repeated and turn out having identical ratio. Ideas regarding sentences are explored. Just as English sentences have verbs,
figures, like the bees’ honeycomb with hexagonal tiles. so do mathematical sentences. In the mathematical sentence;
Harmonic Sequence. In the sequence, the reciprocal of the terms behaved in a 3+4=7
manner like arithmetic sequence the verb is =. If you read the sentence as ‘three plus four is equal to seven, then
Symmetries in Nature it’s easy to hear the verb. Indeed, the equal sign = is one of the most popular
mathematical verb.
Human Body Fibonacci Sequence. This specific sequence was named after an Italian
The human body is one of the pieces of evidence that there is symmetry in mathematician Leonardo Pisano Bigollo (1170 - 1250). He discovered the Connectives
nature. Our body exhibits bilateral symmetry. It can be divided into two sequence while he was studying rabbits. The Fibonacci sequence is a series of The answer is the symbol + is what we called a connective which is used to
identical halves numbers governed by some unusual arithmetic rule. The sequence is connect objects of a given type to get a ‘compound’ object of the same type.
organized in a way a number can be obtained by adding the two previous Here, the numbers 1 and 2 are connected to give the new number 1 + 2
Animal Movement numbers.
The symmetry of motion is present in animal movements. When animals
move, we can see that their movements also Mathematical Sentence
exhibit symmetry Mathematical sentence is the analogue of an English sentence; it is a correct
arrangement of mathematical symbols that states a complete thought. It
Sunflower makes sense to as about the TRUTH of a sentence: Is it true? Is it false? Is it
One of the most interesting things about a sunflower is that it contains both sometimes true/sometimes false?
radial and bilateral symmetry The amazing grandeur of Fibonacci sequence was also discovered in the
structure of Golden rectangle. The golden rectangle is made up of squares Truth of Sentences
Snowflakes whose sizes, surprisingly is also behaving similar to the Fibonacci sequence. Sentences can be true or false. The notion of “truth” (i.e., the property of being
Snowflakes have six-fold radial symmetry. true or false) is a fundamental importance in the mathematical language; this
MATHEMATICAL LANGUAGE AND SYMBOLS will become apparent as you read the book.
Honeycombs/Beehive
Honeycombs or beehives are examples of wallpaper symmetry.
A. Characteristics of Mathematical Language Conventions in Languages
Starfish The language of mathematics makes it easy to express the kinds of thoughts Languages have conventions. In English, for example, it is conventional to
Starfish have a radial fivefold symmetry. Each arm portion of the starfish is that mathematicians like to express. capitalize name (like Israel and Manila). This convention makes it easy for a
identical to each of the other regions. It is: reader to distinguish between a common noun (carol means Christmas song)
1. precise (able to make very fine distinction) and proper noun (Carol i.e. name of a person). Mathematics also has its
2. concise (able to say things briefly); and convention, which help readers distinguish between different types of
Fibonacci in Nature 3. powerful (able to express complex thoughts with relative cases). mathematical expression.
These include the buttercup, columbine, and hibiscus. Aside
from those flowers with five petals, eight-petal flowers like clematis and B. Vocabulary vs. Sentences Expression
delphinium also have the Fibonacci numbers, while ragwort and marigold have Every language has its vocabulary (the words), and its rules for combining An expression is the mathematical analogue of an English noun; it is a correct
thirteen. These numbers are all Fibonacci numbers. Apart from the counts of these words into complete thoughts (the sentences). Mathematics is no arrangement of mathematical symbols used to represent a mathematical
flower petals, the Fibonacci also occurs in nautilus shells with a logarithmic exception. As a first step in discussing the mathematical language, we will object of interest.
spiral growth. Multiple Fibonacci spirals are also present in pineapples and red make a very broad classification between the ‘nouns’ of mathematics (used to
cabbages. The patterns are all consistent and natural. name mathematical objects of interest) and the ‘sentences’ of mathematics E. Conventions in mathematics, some commonly used symbols, its meaning
(which state complete mathematical thoughts)’ and an example

Sequence a) Sets and Logic

Importance of Mathematical Language
Sequence refers to an ordered list of numbers called terms, that may have • Major contributor to overall comprehension
repeated values. The arrangement of these terms is set by a definite rule. The • Vital for the development of Mathematics proficiency
 5. Cardinal Number; n
Cardinal number are numbers that used to measure the number of elements
in a given set. It is just similar in counting the total number of element in a set.

Illustration:
A = { 2, 4, 6, 8 } n = 4
B = { a, c, e }
n=3
F. Translating words into symbol 6. Equal set
• Practical problems seldom, if ever, come in equation form. The job of the Two sets, say A and B, are said to be equal if and only if they have equal
problem solver is to translate the problem from phrases and statements into number of cardinality and the element/s are identical. There is a 1 -1
mathematical expressions and equations, and then to solve the equations. correspondence.
• As problem solvers, our job is made simpler if we are able to translate verbal
phrases to mathematical expressions and if we follow step in solving applied Illustration:
problems. To help us translate from words to symbols, we can use the A = { 1, 2, 3, 4, 5} B = { 3, 5, 2, 4, 1}
Mathematics Dictionary
7. Equivalent set Two sets, say A and B, are said to be equivalent if and only if
they have the exact number of element. There is a 1 – 1 correspondence.
I. SETS AND SUBSETS Illustration:
A = { 1, 2, 3, 4, 5 }
A. The Language of Sets B = { a, b, c, d, e }
Use of the word “set” as a formal mathematical term was introduced in 1879
by Georg Cantor. For most mathematical purposes we can think of a set 8. Universal set The universal set U is the set of all elements under discussion.
intuitively, as Cantor did, simply as a collection of elements.
Illustration:
Terminologies of Sets A set of an English alphabet
b) Basic Operations and Relational Symbols U = {a, b, c, d, …, z}
1. Unit Set
Unit set is a set that contains only one element. 9. Joint Sets
Two sets, say A and B, are said to be joint sets if and only if they have common
Illustration: element/s.
A = { 1 }; B = { c }; C = { banana }
A = { 1, 2, 3}B = { 2, 4, 6 }
2. Empty set or Null set;  Here, sets A and B are joint set since they have common element such as 2.
Empty or null set is a set that has no element.
10.Disjoint Sets
Two sets, say A and B, are said to be disjoint if and only if they are mutually
Illustration: exclusive or if they don’t have common element/s.
A={}
A set of seven yellow carabaos A = { 1, 2, 3}B = { 4, 6, 8 }

3. Finite set
A finite set is a set that the elements in a given set is countable. B. Two ways of Describing a Set

Illustration: 1. Roster or Tabular Method

A = { 1, 2, 3, 4, 5, 6 } It is done by listing or tabulating the elements of the set.
B = { a, b, c, d }
2. Rule or Set-builder Method
4. Infinite set It is done by stating or describing the common characteristics of the elements
An infinite set is a set that elements in a given set has no end or not of the set. We use the notation A = { x / x … }
countable.
Illustration:
c) Set of Numbers Illustration: a. A = { 1, 2, 3, 4, 5 }
A set of counting numbers A = {x | x is a counting number from 1 to 5}
A = { …-2, -1, 0, 1, 2, 3, 4, … } A = { x | x  N, x < 6}
b. B = { a, b, c, d, …, z } A  B = { x | x  A or x  B } type of diagram is used in scientific and engineering presentations, in
B = {x | x  English alphabet} Example 1: If A = {1, 2, 3} and B = {4, 5} , then theoretical mathematics, in computer applications, and in statistics
B = { x | x is an English alphabet} A  B = {1, 2, 3, 4, 5} .
Example 2: If A = {1, 2, 3} and B = {1, 2, 4, 5} , then
C. Subsets A  B = {1, 2, 3, 4, 5} .
A subset, A  B, means that every element of A is also an element of B. Note that elements are not repeated in a set.

If x  A, then x  B. In particular, every set is a subset of itself, A  A. 2. Intersection of Sets

A subset is called a proper subset, A is a proper subset The intersection of sets A and B, denoted by A  B , is the set defined as :
of B, if A  B and there is at least one element of B that is not in A: A  B = { x | x  A and x  B }
If x  A, then x  B and there is an element b such that b  Example 1: If A = {1, 2, 3} and B = {1, 2, 4, 5}
B and b  A. then A  B = {1, 2} .
Example 2: If A = {1, 2, 3} and B = {4, 5}
NOTE1: The empty set. or {} has no elements and is a subset of every set for then A  B = 
every set A, A  A.
3. Difference of Sets
The number of subsets of a given set is given by 2n , where n is the number of The difference of sets A from B , denoted by A - B , is the set
elements of the given set. defined as II. FUNCTIONS AND RELATIONS
A - B = { x | x  A and x  B }
Illustration: What are the elements (ordered pair) in A x B, based on the given conditions, that
How many subsets are there in a set Example 1: If A = {1, 2, 3} and B = {1, 2, 4, 5} are related?
Perhaps your answer would be:
A = {1, 2, 3 }? List down all the subsets of set A. Number of subsets = 2n = 23 = then A - B = {3} .
{ (1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}
8 subsets Example 2: If A = {1, 2, 3} and B = {4, 5} ,
Observe that knowing which ordered pairs lie in this set is equivalent to knowing
With one element then A - B = {1, 2, 3} . which elements are related to which. The relation can be therefore be thought of the
{1};{2};{3} Example : 3 If A = {a, b, c, d } and B = {a, c, e } , totality of ordered pairs whose elements are related by the given condition. The
With two elements then A - B = {b, d } . formal mathematical definition of relation, based on this idea, was introduced by the
{ 1, 2 } ; { 1, 3 }; { 2, 3 } Note that in general A - B  B - A American mathematicians and logician C.S. Peirce in the nineteenth century.
With three elements
{ 1, 2, 3 } 4. Compliment of Set
With no elements For a set A, the difference U - A , where U is the universe, is called the What is a relation?
{} complement of A and it is denoted by Ac . Thus Ac is the set of everything that 1. A relation from set X to Y is the set of ordered pairs of real numbers (x, y) such
that to each element x of the set X there corresponds at least one element of the
is not in A.
set Y.
D. Ordered Pair Example: Let U = { a, e, i, o, u } and A = { a, e } 2. Let A and B sets. A relation R from A to B is a subset of A x B. Given an ordered pair
Given elements a and b, the symbol (a, b) denotes the ordered pair consisting then Ac = { i, o u } (x, y) in A x B, x is related to y by R, written x R y, if and only if, (x, y) is in R. The set A is
of a and b together with the specification that “a” is the first element of the called the domain of R and the set B is called its co domain.
pair and “b” is the second element. Two ordered pairs (a,b) and (c,d) are equal 5. Cartesian Product
iff a = c and b = d. Symbolically; Given sets A and B, the Cartesian product of A and B, denoted by A x B and
(a, b) = (c, d) means that a = c and b = d read as “A cross B”, is the set of all ordered pair (a,b) where a is in A and b is in C. ARROW DIAGRAM OF A RELATION
B. Symbolically: Suppose R is a relation from a set A to a set B. The arrow diagram for R is obtained
Illustration: A x B = {(a, b) | a  A and b  B} as follows:
a) If (a, b) = (3, 2), what would be the value of a and b. Note that A x B is not equal to B x A. 1. Represent the elements of A as a points in one region and the elements of B as
points in another region.
Here, by definition that two ordered pairs (a,b) and (c,d) are equal iff a = c
2. For each x in A and y in B, draw an arrow from x to y, and only if, x is related
and b = d. Illustration: to y by R. Symbolically:
Hence, a = 3 and b = 2. If A = { 1, 2} and B = {a, b}, what is A x B?
b) Find x and y if (4x + 3, y) = (3x + 5, – 2). A x B = {(1,a), (1, b), (2, a), (2, b)}. How many elements in a A x B? Draw an arrow from x to y
Solution: Example 1: Let A = {1, 2, 3} and B = {a, b}. Then If and only if, x R y
Since (4x + 3, y) = (3x + 5, – 2), so A x B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)} If and only if, (x, y)  R.
4x + 3 = 3x + 5
Solving for x, we got x = 2 and obviously y = – 2. Example 2: For the same A and B as in Example 1, Example:
B x A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} . 1. Given a relation {(1, 2),(0, 1),(3, 4),(2, 1),(0, −2)}. Illustrate the given relation
into an arrow diagram.
E. OPERATION ON SETS
Sets can be combined in a number of different ways to produce another set. Venn Diagram
Here are the basic operations on sets. A Venn diagram is an illustration of the relationships between and among sets,
groups of objects that share something in common. Usually, Venn diagrams are
1. Union of Sets used to depict set intersections (denoted by an upside-down letter U). This
The union of sets A and B, denoted by A  B , is the set defined as:
 in which no two distinct ordered pairs have the same first component. Similar to a
relation,the values of x is called the domain of the function and the set of all resulting
value of y is called the range or co-domain of the function.
A function F from a set A to a set B is a relation with domain and co-domain
B that satisfies the following two properties:
1. For every element x in A, there is an element y in B such that (x,y) F.
2. For all elements x in A and y and z in B,
If (x,y)  F and (x,z) F, then y = z

These two properties; (1) and (2) can be stated less formally as follows:
1. Every element of A is the first element of an ordered pair of F.
D. PROPERTIES OF A RELATION 2. No two distinct ordered pairs in F have the same first element.
When a relation R is defined from a set A into the same set A, the three properties are • Is a function a relation? Focus on the x-coordinates, when given a relation.
very useful such as reflexive, symmetric and the transitive. • If the set of ordered pairs have different x-coordinates, it is a function.
• If the set of ordered pairs have same x-coordinates, it is NOT a function but it
A. Reflexive could be said a relation.
A relation R on A is said to be reflexive if every element of A is related to itself. In
notation, a R a for all a Note:
 A. Examples of reflexive relations include: a) Y-coordinates have no bearing in determining functions
_ "is equal to" (equality) b) Function is a relation but relation could not be said as function.
_ "is a subset of" (set inclusion)
_ "is less than or equal to" and "is greater than or equal to" (inequality)
_ "divides" (divisibility). Function Notations:
An example of a non reflexive relation is the relation "is the father of" on a set of The symbol f(x) means function of x and it is read as “f of x.” Thus, the equation y = 2x
people since no person is the father of themself. When looking at an arrow diagram, a + 1 could be written in a form of f(x) = 2x + 1 meaning y = f(x). It can be stated that y is
relation is reflexive if every element of A has an arrow pointing to itself. For example, a function of x.
the relation in a given figure below is a reflexive relation. Let us say we have a function in a form of f(x) = 3x – 1. If we replace x = 1,
this could be written as f(1) = 3(1) – 1. The notation f(1) only means that we substitute
B. Symmetric the value of x = 1 resulting the function value. Thus
A relation R on A is symmetric if given a R b then b R a. For example, "is married to" is
a symmetric relation, while, "is less than" is not. The relation "is the sister of" is not f(x) = 3x – 1; let x = 1
symmetric on a set that contains a brother and sister but would be symmetric on a set f(1) = 3(1) – 1 = 3 – 1 = 2.
of females. The arrow diagram of a symmetric relation has the property that whenever Another illustration is given a function g(x) = x2 – 3 and let x = -2, then g(-2) = (-2)2 – 3
there is a directed arrow from a to b, there is also a directed arrow from b to a. =1

C. Transitive
A relation R on A is transitive if given a R b and b R c then a R c.
Examples of reflexive relations include:
_ "is equal to" (equality)
_ "is a subset of" (set inclusion)
_ "is less than or equal to" and "is greater than or equal to" (inequality)
_ "divides" (divisibility).
On the other hand, "is the mother of" is not a transitive relation, because if Maria is the
mother of Josefa, and Josefa is the mother of Juana, then Maria is not the mother of
Juana. The arrow diagram of a transitive relation has the property that whenever there
are directed arrows from a to b and from b to c then there is also a directed arrow
from a to c:
A relation that is refexive, symmetric, and transitive is called an equivalence
relation on A.
Examples of equivalence relations include:
_ The equality ("=") relation between real numbers or sets.
_ The relation "is similar to" on the set of all triangles.
_ The relation "has the same birthday as" on the set of all human beings.
On the other hand, the relation "  " is not an equivalence relation on the set of all
subsets of a set A since this relation is not symmetric.

E. WHAT IS A FUNCTION?
A function is a relation in which every input is paired with exactly one output.
A function from set X to Y is the set of ordered pairs of real numbers (x, y)