Chapter 1

Draw the greek letter phi

(illustrate)

Reviw subset and proper subset

(study those two and symbols)

Different Mathematical Conventions:

1. Mathematics has its own brand of teachnical terms. 2. Mathematical statements also have its own taxonomy 3.Mathematics also have mathematical jargon 4. The vocabulary of Mathematics also has visual elements 5. The mathematical notation has its own grammar and does not dependant on a specific natural language 6. Mathematical expressions containing a symbolic verb are generally treated as clauses im sentences or as a complete sentences and are punctuated as such by mathematicians.

Two ways of Representing a set

1. Roster method (Tabulation method ) 2. Rule Method (set builder notation )

Branches of Mathematics

Algebra, Arithmetic , Calculus, Geometry , Trigonometry

Types of Symmetry Patterns

Bilateral Symmetry, Radial Symmetry

It is a symmetry in which the left and right sides of the organism is canbe divided into approximately mirror image of each other along the midline.

Bilateral or (Vertical) Symmetry

It is the number of members of the set. ex. D= {a, e, i, o , u} answer: n (D)=5

Cardinal Numbers

denoted as " A x B" example: A={1, 2} B= {2,4} answer: AxB={(1,2), (1,4), (2,2), (2,4), (2,1), (4,1), (4,2)}

Cartesian product

x in U such that x is not in A Complement of A ..denoted as"A' " example: A={a,b,c} U={a,b,c,d,e,f,g,h} answer: A'={d,e,f,g,h}

Complement of set

Simpest possible single exposition at the price of additional terminology and machinery.

Concise

Types of Radial Symmetry

Cyclic or dihedral

x in U such that x is in A and x is not in B denoted by "A-B" example: A={a,b,c} B={c,d,e} answer: A-B={a,b}

Difference of set

two sets have no elements in common. Also called as non-intersecting set.

Disjoint set

Golden ratio was first called as____in 1500.

Divine Proportion

the members or objects of the set and is denoted by "E" (not literally E like this.

Elements

It is a unique set with no elements and also called as the Null set. Denoted by {0} or { }. ex: L={x/x is an integer less than 2 but greater than 1}.

Empty set

Subset of any set

Empty set or { }

It is a sets that re equal if and only if every element of A is in B and every element of B is in A

Equal set

Two set have the ame number of elements and is denoted by (~)

Equivalent set

First to give the definition of the golden ratio as "a dividing line in the extreme and mean ratio" in his book the "elements"

Euclid

denoted as A U (may equals sa baba) A={1,2,3,4,5} B={3,4,5,6,7} answer: A U(equal sa baba) B ={1,2,6,7}

Exlusive uniom /symmetric difference

Series of numbers in which the next number is found by adding up the two numbers before it.

Fibonacci sequence

Is a set whose elements are limited or countable and the last element can be identified. ex. D={12,14,13,14,15}

Finite Set

Types of set

Finite set, Infinite set, Unit set, Empty set , Universal set, Subset, equal set, equivalent set, disjoint set

Example of Golden Ratio in nature

Flower petals, Faces, Body Parts , Seed heads, Fruits vegetable trees, Shells , spiral Galaxies, Hurricanes

A never ending pattern foun in nature. A curve or geometric figure each part of which has the same statistical characteristics as the whole

Fractals

Is a special kind of reation helps visualize relationship in terms of graphs and makes it easier to different behavior of variables.

Function

Set has become a fundamental theory in 1870. And is introduced by________

Gerge Cantor

Is a set whose elements are unlimited or countable and the last element cannot be specified. ex: G={....-3,-2,-1,0,1,2,3.....}

Infinite set

x in U such that x is in A and x is in B Ex: A={a,b,c} B={c, d, e} answer: A (inverted U) B={c}

Intersection of set

Serve as a toll for teaching mathematical concepts. Serve as a major pedagogical tool to understand how, what and why things are said.

Language

Unordered colections of obejects called elements or members of the set.

Language of sets

Fibonacci Sequence is name after?

Leonardo Pisano or (Fibonacci)

Fibonacci numbers were first introduced in 12012 in what book?

Liber Abacci (Book of Calculation )

It is a fact, name, notation, or usage which is generally agreed upon by mathematicians.

Mathematical Conventions

System used to communicate mathematical ideas

Mathematical Language

Study of pattern and structure / tool to quantify , organize and control our worl, predict phenomena and make life easier for us.

Mathematics

The science of numbers and their operations, interrelations, combinations.

Mathematics

The dazzling display of avian behavior of some living organisms that make shape formations when they move as a group like when flocks of starlings create incredible aerobotic displays. It is the word that perfectly describes the rustle of thouands of pair of wings.

Murmuration

Vital Clues that govern natural processes

Nature patterns

Visible regularities of form found in natural world and can also be seen in the universe

Patterns in Nature

P (S) =2^n example: A={c, d, e} n=3 P (A) =2^3=8 P (A)= {{c}, {d} , {e}, {c,d}, {c, e}, {d, e}, {c, d, e} {0} }

Power sets

Ways of expressing complex thoughts with relative ease

Powerful

In Mathematics it is a culture of being correct all the time

Precision

It is a symmetry around a fixed point known as the center.

Radial symmetry (rotational symmetry)

a R b A={a,b} B={c,d} R={(a,c), (a,d), (b,c), (b,d)}

Relation

Spiral is first describe by:_____ and was later investigated by:_______

Rene Descartes Jacob Bernoulli

It is when the elements of the set are enumerated and separated by a comma. ex. A={x/x is a positive integer less than 10} answer: A={1,2,3,4,5,6,7,8,9}

Roster Method ( Tabulation Method )

It is used to described the elements or members of the set. ex. F= {12} answer: F={x/x=12}

Rule method (set builder notation)

It is a process in a Fractal Symmetry in which the exact shape is replicated.

Self Similarity

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

Set

Subset of itself

Set

Branch of Mathematics that studies sets or the mathematical science of finite.

Set theory

It is a curved pattern that focuses on a certain point and a series of circular shapes that revolve around it.

Spiral

It is a sense of harmonious and beautiful proportion of balance or an object ys invariant to any various transformations (reflections, rotation or scaling)

Symmetry

What are the types of pattern?

Symmetry , Fractals , Spiral

Natural Regularitues of nature

Symmetry , Trees, Foams, stripes, Fractals, Meanders, Tessellations, Spots, Spirals, Waves, Cracks

What are the Operations on set?

Union of set, Intersection of set, Complement of set, Difference of Set, Cartesian Product, power sets, Exclusive union

x in U such that x is in A or x is in B Sample: A={a,b,c} B={c, d, e} answer: A U B={a, b, c, d, e}

Union set

Is a set with only one element, it is also called "singleton". ex: J={January}

Unit set

Is the totality of the set, all sets under investigation in any application of set theory are assumed to be contained in some large fixed set and is denoted by "U" example: A={a,b , c} B={d, e, f} answer: U={a,b,c,d,e,f}

Universal set

Formula for Fibonacci Sequence

X(sub)n=X (sub)n-1+X (sub)n-2

In Timaeus Plato describes the Platonic Solid. What are those?

tetrahedron, cube, octahedron, dodecahedron, icosahedron

Given {(1,2) , (4,5) , (5,6)} Domain: {1,4,5} image (range): {2,5,6}

domain and range

correct arrangement of mathematical symbols used to represen the object of interest, it does not contain a complete thought and it cannot be determined if it os true or false.

expression

us a relation in which for each value of the first component of the ordered pairs, there is exactly one value of the second component

function

It is a special number approximately equal to 1.618, an irrational number

golden ratio

function-hindi nuulit si x relation- nauulit si x

pano malaman kung relation or function

Characteristics of mathematical language

precise, concise, powerful

set of all the images of the elements of the domain

range

Set of ordered pairs

relation

correct arrangement of mathematical symbols that states a complete thought and can be determined if it os true or false.

sentence

stude venn diagram

study