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Mathematics Is the study of pattern and structure Fundamental to the physical and biological sciences, engineering and information technology, to economics, and increasingly to the social sciences Is a useful way to think about nature and our world Is a tool to qualify, organize, and control our world, predict, phenomena, and make life easier for us. Where is mathematics Many patterns and occurrences exist in nature, in our world, in our life. Mathematics helps makes sense of these patterns and occurrences. WHAT ROLE DOES MATHEMATICS PLAY IN OUR WORLD? 

Mathematics helps organize patterns and regularities in our world.

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Mathematics helps predict the behavior of nature and phenomena in the world.

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Mathematics helps control nature and occurrences in the world for our own ends.

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Mathematics has numerous applications in the world making it indispensable.

PATTERNS AND NUMBERS IN NATURE AND THE WORLD 

Patterns in nature are visible regularities of form found in the natural world and can also be seen in the universe.

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Nature patterns which are not just to be admired, they are vital clues to the rules that govern natural processes.

Check out some examples of these patterns that you may be able to spot the moment you decided to go for a walk. 1. Patterns can be observed even in stars that move in circles across the sky each day. 2. The weather season cycle each year. All snowflakes contain six-fold symmetry which no two are exactly the same. 3. Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lionfish, yellow boxfish, and angelfish. These animals and fish stripes and spots attest to mathematical regularities in biological growth and form. 4. Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lionfish, yellow boxfish, and angelfish. These animals and fish stripes and spots attest to mathematical regularities in biological growth and form. 5. Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lionfish, yellow boxfish, and angelfish. These animals and fish stripes and spots attest to mathematical

regularities in biological growth and form. 6. Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lionfish, yellow boxfish, and angelfish. These animals and fish stripes and spots attest to mathematical regularities in biological growth and form.

PATTERNS AND REGULARITIES SYMMETRY – a pattern with a sense of harmonious and beautiful proportion of balance or an object is invariant to any various transformations examples are reflection, rotation or scaling 1. Bilateral Symmetry - is symmetry in which the left and right sides of the organism can be divided into approximately mirror images of each other along the midline. This exists in living things like insects, animals, plants, flowers, and others. Animals can further be classified as either cyclic or dihedral.Plants on the other hand often have radial or rotational symmetry, as to flowers and some group of animals.There is also what we call a five-fold symmetry which is found in the echinoderms, the group which includes starfish (dihedral-D5 symmetry), sea urchins, and sea lilies.Radial symmetry suits organisms like sea anemones whose adults do not move and jellyfish(dihedral-D4 symmetry). Radial symmetry is also evident in different kinds of flowers. FRACTALS – a pattern with a curve or geometric figure, each part of which has the same statistical character as the whole. A fractal is a never-ending pattern found in nature. The exact same shape is replicated in a process called “self- similarity.” The pattern repeats itself over and over again at different scales. For example, a tree grows by repetitive branching. This same kind of branching can be seen in lightning bolts and the veins in your body SPIRALS – are a curved pattern that focuses on a center point of a series of circular shapes that revolve around it. A logarithmic spiral or growth spiral is a self-similar spiral curve which often appears in nature. It was first described by Rene Descartes and was later investigated by Jacob Bernoulli. Examples of spirals are pine cones, pineapples, hurricanes. FIBONACCI SEQUENCE Is a series of numbers where a number is found by adding up the two numbers before it. Starting with 0 and 1, the sequence goes 0,1,1,2,3,5,8,13,21,34, and so forth. Mathematically it can be expressed as . This is named after Fibonacci, who is also known as Leonardo of Pisa or Leonardo Pisano. Fibonacci numbers were first introduced in his Liber Abbaci (Book of Calculation) in 1202.

he sequence encountered in the rabbit problem 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …. is called the Fibonacci sequence. Each term in the sequence is called the Fibonacci numbers. Fibonacci Sequence can also be seen in shape and nature.

An example of this is shown in the so-called Golden Rectangle... GOLDEN RATION It was first called the DIVINE PROPORTION in early 1500s in Leonardo da Vinci’s work Mathematically two quantities are in the Golden ratio if (a+b) divided by A is equal to a divided by b which is equal to 1.618033987…. and represented by (phi), provided that A is greater than B. Examples: 1. If a = 3 and b = 2 then a/b = 1.5 2. If a = 5 and b = 3 then a/b = 1.666666... 3. If a = 8 and b = 5 then a/b = 1.6 4. if a = 13 and b = 8 then a/b= 1.625 5. If a = 21 and b = 13 then a/b = 1.615384615... In the same manner, the golden ratio can also be noticed in Arts let us name a few... 1. The exterior dimension of the Pathernon in Athens, Greece embodies the golden ratio. 2. In Timaeus Plato describes five possible regular solids that relate to the golden ratio which is now known as Platonic Solid. 3. Euclid was the first to give a definition of the golden ratio as a dividing line in the extreme and mean ratio in his book the Elements. 4. Leonardo da Vinci used the golden ratio to define the fundamental portions of his works. He incorporated the golden ratio in his paintings such as" The Last Supper", "Monalisa" and "St. Jerome in the Wilderness". 5. Michael Angelo di Lodovico Simon was considered the greatest living artist of his time. He used the golden ratio in his painting " The Creation of Adam ". 6. Raffaello Sanzio da Urbino was a painter and an architect from a renaissance. In his painting "The School of Athens", the division between the figures in the painting and their proportions are distributed using the golden ratio.

GOLDEN RATION IN ARCHITECTURE 1. Great Pyramid of Giza built 4700 Bc in Ahmes Papyrus of Egypt , the ratio its base to the height is roughly 1.5717 which is close to the golden ratio 2. Notre Dame is a Gothic Cathedral in Paris. 3. Taj Mahal is found in India and used the golden ratio in its construction and was completed in 1648 4. Cathedral of Our Lady of Chartres in Paris, France which also exhibits the golden ratio 5. The United Nation Building the window configuration reveal the golden proportion. MATHEMATICAL LANGUAGE Is the system used to communicate mathematical ideas This language consists of some natural language using technical terms

(mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas. -The mathematical notation used for formulas has its grammar and shared by mathematicians anywhere in the globe.

Mathematical language must be precise, concise, and powerful, these must be its characteristics.

FIBONACCI NUMBERS IN NATURE These flower petals exhibit the Fibonacci number, white calla lily contains 1 petal, euphorbia contains 2 petals, trillium contains 3 petals, columbine contains 5 petals, bloodroot contains 8 petals, black-eyed Susan contains 13 petals, Shasta daisies 21 petals, field daisies contains 34 petals and other types of daisies contain 55 and 89 petals. FIBONACCI NUMBERS IN NATURE 

These flower petals exhibit the Fibonacci number, white calla lily contains 1 petal, euphorbia contains 2 petals, trillium contains 3 petals, columbine contains 5 petals, bloodroot contains 8 petals, black-eyed Susan contains 13 petals, Shasta daisies 21 petals, field daisies contains 34 petals and other types of daisies contain 55 and 89 petals

CHARACTERISTIC OF MATHEMATICAL 1. Precision – in mathematics is a culture of being correct all the time. Definition and limits should be distinct. Mathematical ideas are being developed informally and being done more formally, with necessary and sufficient conditions stated upfront and restricting the discussion to a particular class of objects. 2. Concise in mathematics must show simplicity. Being concise is a strong part of the culture in mathematical language. Mathematicians desire the simplest possible single exposition. 3. Mathematical language must also be powerful. It is a way of expressing complex thoughts with relative ease. The abstraction in mathematics is the desire to unify diverse instances under a single conceptual framework and allows easier penetration of the subject and the development of more powerful methods. EXPRESSION An expression (or mathematical expression) is a finite combination of symbols that is well-defined according to rules that depend on the context. The symbols can designate numbers, variables, operations, functions, brackets, punctuation, and groupings to help determine the order of operations and other aspects of mathematical syntax. An expression is a correct arrangement of mathematical symbols used to represent the object of interest, it does not contain a complete thought, and it cannot determine if it is true or false. Some types of expressions are numbers, sets, and functions. SENTENCES A sentence (or mathematical sentence) makes a statement about two expressions, either using numbers, variables, or a combination of both. A mathematical sentence can also use symbols or words like equals, greater than, or less than.

A mathematical sentence is a correct arrangement of mathematical symbols that states a complete thought and can be determined whether it’s true, false, and sometimes true/sometimes false.

CONVENTIONS IN THE MATHEMATICAL LANGUAGE Mathematical languages have conventions and it helps individuals distinguish between different types of mathematical expressions. The mathematical convention is a fact, name, notation, or usage which is generally agreed upon by mathematicians. Mathematicians abide by conventions to be able to understand what they write without constantly having to redefine basic terms. 1. Mathematics has its brand of technical terms. – a word in general usage has a different and specific meaning within mathematics. Ex. Group, ring, field, term, factor, etc. 2. Mathematical statements also have their taxonomy. Ex. Axiom, conjecture, theorems, lemma, and corollaries. 3. Mathematics also has Mathematical jargon- mathematical phrases used with specific meanings. Ex. “If and only if”, “necessary and sufficient” and “without loss of generality.” 4. The vocabulary of mathematics also has visual elements. Ex. Used informally in blackboards and formally in books and researches which serve to display schematic information so easily. 5. The mathematical notation has its grammar and does not dependent on a specific natural language. Ex. Latin alphabet used for simple variables and parameters. 6. Mathematical expressions containing a symbolic verb are generally treated as clauses in sentences or as a complete sentence and are punctuated as such by mathematicians. Ex. Equal ( = ) , Less than ( < ) , Greater than ( > ) , Addition (+), Subtraction (-) , Multiplication (x), infinity ( ∞∞), for all ( ∀∀) , there exists (∋∋ ), element (∈∈ ) , implies (⟶⟶ ),if and only if (⟷⟷ ), therefore ( ∴∴) etc.

Set Theory is the branch of mathematics that studies sets or the mathematical science of the infinite. 

The study of sets has become a fundamental theory in 1870.

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Introduced by George Cantor (German Mathematician.)

SET 

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is a collection of well defined objects. usually denoted by capital letters of the alphabet and its members are enclosed with brackets.

Elements – are the members or objects of the set which is denoted by a symbol (∈ ).

Example of a set: A-{ x/x is a set of letters from the word Pneumonia} This is read as A is the set of all x such that x is a set of letters from the word Pneumonia. The elements of this set are a, e, i, m, n, o, p, u. TWO WAYS OF REPRESENTING A SET 1. ) Roster Method (Tabulation Method) – when the elements of the set are enumerated and separated by a comma. Ex.

A={ 23, 25, 27}

Write the following Set in Roster Method 1. A={x/x is a positive integer less than 10} A= { 1, 2, 3, 4, 5, 6, 7, 8, 9} 2. B={x/x is a month in the calendar} B= { Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sept, Oct, Nov, Dec} 3. C={x/x is an integer , 1< x < 8 } C={2, 3, 4, 5, 6, 7} 2.) Rule Method ( Set builder notation) - used to describe the elements or members of the set using their common characteristics. Ex. B= {x/x is a set of professors from the Math and Physics Department} Write the following Set in Rule Method 1. D={ a, e, i, o, u} D= {x/x are vowel letters from the alphabet} 2. E={ 4, 6, 8, 10, 12, 14, 18, 20} E={x/x is even number from 4 to 20} 3. F= {12} F=[x/x is equal to 12} Cardinal Number- this refers to the number of elements in a given set. The cardinality of a set is given by n(A). Examples: Identify the Cardinality of the given sets. 1. A= { 1, 2, 3, 4, 5, 6, 7, 8, 9} n(A)=9

2. B={x/x is a month in the calendar} n(B)=12 3.C={x/x is an integer , 1< x < 8 }

n(C)= 6 4. D={ a, e, i, o, u} n(D)= 5 5. E={ 4, 6, 8, 10, 12, 14, 18, 20} n(E) = 8 6. F= {12} n(F) = 1 TYPES OF SET 1. Finite Set – is a set whose elements are limited or countable and the last element can be identified. 2. Infinite Set – is a set whose elements are unlimited or uncountable and the last element cannot be specified. 3. Unit Set – is a set with only one element, it is also called a singleton. 4. Empty Set – a unique set with no elements and also called as the Null Set. It is denoted by { }. 5. Universal Set – the totality of the set, all sets under investigation in any application of set theory are assumed to be contained in some largely fixed set and is denoted by U. 6. Subset - if A and B are set, A is called a subset of B, written A ⊆ B, if and only if, every element of A is also an element of B. A is a proper subset of B, written A ⊂ B, if and only if, every element of A is in B but there is at least one element of B that is not in A. 7. Equal Set - two sets are equal if and only if, every element of A is in B and every element of B is in A. 8. Equivalent Set - two sets are equivalent if they have the same number of elements and it is denoted by (~). 9. Disjoint set - two sets that do not have the same elements. This is also known as a non-intersecting set. Given: 1. A = { a , b , c } = finite set 2. B = { a , b , c, d, e } = finite set 3. C ={ a , b , c, e, d .... } =infinite set 4. D = { } = empty set/ null set 5. E = { bat } = unit sets

6. U = { a , b , c, d, e, bat,} = universal set Examples: Which of these are subsets, equal sets, equivalent sets, and disjoint set.

Given: A={a,b,c} B = { a , b , c, d, e } C = { a , b , c, e, d } D = { f , g, h , i } E = { 1, 2, 3, 4 } F={1,5} U = { a , b , c, d, e, f, g, h, i, j, 1, 2, 3, 4, 5} Based on the definitions of each set we can have the following:  

B is a subset of C, mathematically B⊆CB⊆CBB ⊆ CBB ⊆ C A is a proper subset of B, mathematically A⊂BA⊂BAA ⊂ BAA ⊂ B

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B and C are equal set, they have the same elements.

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D is equivalent to E, mathematically D∼ED∼EDD ∼ EDD ∼ E, these sets have the same number of elements.

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B is equivalent to C, mathematically B∼CB∼CBB ∼ CBB ∼ C, these sets have the same number of elements.

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D and E are disjoint sets.

OPERATIONS ON SETS: 1. UNION OF SET- the union of A and B, denoted by A∪ B, is the set of all elements in x in U such that x is in A or x is in B. 2. INTERSECTION OF SET - the intersection of A and B, denoted by A ∩ B, is the set of all elements in x in U such that x is in A and x is in B. Given: A= { a, b, c } B = { c, d, e } C = { f, g } D = { f, g, h, i} Let us answer the set of examples: a. A ∪ B = { a , b, c, d, e}

d. A ∩ B = { c ]

b. C ∪ D = { f , g, h, i }

e. C ∩ D = { f, g }

c. B ∪ C = { c, d, e, f, g }

f. B ∩ C

={

}

3. COMPLEMENT OF SET- The complement of a set or absolute complement A,

denoted by A , is the set of all elements in x in U such that x is not in A. Given: A= { a, b, c } B= { c, d, e }

U = { a, b, c, d, e, f, g, h } Examples; A' = { d, e, f, g, h} B' = { a , b, f, g, h } (A' ∩ B' ) ={ f, g, h} 4. DIFFERENCE OF SET - The difference of A and B ( or relative complement of B with respect to A) , denoted by A - B, is the set of all elements x in U such that x is in A and x is not in B. Given: A={ a, b, c } B = { c, d, e } C = { f, g } D = { f, g, h, i} U = { a, b, c, d, e, f, g, h , i} Find the following: a. A - B = { a, b} b. C - D = { } c. B - C = { c, d, e

CARTESIAN PRODUCT - The Cartesian product of set A and B , written as A x B is the set of all possible ordered pairs with first element from A and second element from B: A x B = {(a, b )/ a ∈ A and b ∈ B } Example: Let A = { 2, 3, 5 } and B = { 7 , 8 } Find each set ; a. A x B ={ ( 2, 7), ( 2, 8), ( 3, 7 ), ( 3, 8 ), ( 5, 7 ), ( 5, 8 ) } b. B x A = { ( 7, 2 ), ( 7, 3 ), ( 7, 5 ), ( 8, 2 ) , ( 8, 3 ), ( 8, 5 )} c. A x A ={ ( 2, 2), ( 2, 3 ), ( 2, 5), ( 3, 2), ( 3, 3), ( 3, 5), ( 5, 2 ), ( 5, 3 ), ( 5, 5) } Venn Diagram- is an illustration that uses circles to show the relationships among things or finite groups of things. Circles that overlap have a commonality while circles that do not overlap do not share those traits. The circles are being placed inside a box, where the box represents the universal set and the shaded inside of a circle represents the subset of a universal set. Sometimes we will use the Venn Diagram for a particular set whose elements are known, the elements should be listed accordingly.

Given: U = { 1, 2, 3, 4, 5 } and A = { 2 , 4 }

APPLICATION OF VENN DIAGRAM If 380 students are taking courses: 215 taking Biology, 173 taking Physics, 182 taking chemistry. 72 taking Biology and Physics, 90 taking Biology and Chemistry, 60 taking Physics and Chemistry Solution: Let A = Biology ( 215) B = Physics ( 173) C = Chemistry ( 182 ) A⋂B = 72 A⋂C = 90 The intersection of the three courses will be label as x. Then... A⋂B = 72 - x A⋂C = 90 - x B⋂C = 60 - x Let us solve for the equation of A: A = 215 - [ ( 72-x) +x + (90-x)] A = 215 - 162 + x A= 53 + x Let us proceed with B: B =173 - [ ( 72 - x) + x + ( 60 - x )] B = 173 - (132 - x) B = 41 + x Now let us have the C: C =182 - [ ( 90 - x) + x + ( 60 - x )] C = 182 - (90 - x + 60) C = 32 + x Now let us find for x... 53 + x + 72 - x + 41 - x + 90 - x + 60 - x + 32 - x + x

348 + x x = 32

380

The value of x will be substituted to find the answer to the Venn diagram...

A= 53 + 32 = 85 B = 41 + 32 = 73 C = 32 + 32 = 64 A⋂B = 72 - 32 = 40 A⋂C = 90 - 32 = 58 B⋂C = 60 - 32 = 6

Language of Relations

Definition:  

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A relation is a set of ordered pairs. If x and y are elements of these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x and i...