book - 1st composition

8/3/2019 Book - 1st Composition

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Chapter 1: Sets and Functions

Section 1: Introduction to Sets

Definition 1.1.1: A set,X, is a collection ofwhatever(without anyrepeated elements) written as a list between a { and a }.

Definition 1.1.2: A source setis a set that contains all elements

contained in the sets of which it is the source (the unsaid sets).

Axiom 1: There exists source set, U, which is a collection ofwhatever.

Example 1.1.1:

Definition 1.1.3:A is a subsetofB (AB) if and only if all elements inA

are contained in B (for any a A, a B).

Axiom 2: There exists a subset,A, of a set, U, which is defined in

general as:

1

Example 1.1.2: Using the same U, letAUand BUsuch that:

1 The : means where, and the aU is assumed

1

2

U5

7 4

xiwloe

not an apple

5

7 4

1 Uxiwloe

not an apple

2

A

B

love

love


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Theorem 1.1.1: For any given source set, U, there exists an empty set,

, and it is a subset ofU.

Proof: LetUbe any given set. Then there exists a subset,A, ofUdefined as: . Suppose there is an element, aA.

We can say thataUsinceAU. However, aUby how weve

definedA, a contradiction, which means thatA has no elements, or

A = { } = , the symbol for the empty set.

We could define the intersection and the union of two sets right now,

but lets move to the next section so that we can define them using

operations.


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Section 2: Introduction to Functions

For the following functions choose some Uto be a source set.

Definition 1.2.1: LetUbe a source set and letM, NU. A function,y, isdefined as a mapping from a setMto a setN, and is denoted ,

where .

Axiom 3: There exists a function.

Definition 1.2.2: An elementmMis a single elementif and only if it is

not a set of more than one element, meaning at its base is a single

element that is not a set.

Theorem 1.2.1: Ifm is a set of a set to the nth degree ofm1, a single

element that is not a set, then m = {{{m1}}} is a single element.

Proof: Letmn+1 = {mn} and letm1 be a single element. n=1: Ifm2 =

{m1}, then m2 is a set of one element and is thus a regular element.

n=2: {{m1}}={m2}=m3 which is also a set of a set of a single

element. n=k: Suppose there are kmany sets of {} around m1,

denoted mk+1 = {{{m1}}}, and this was also determined to be

one thing, a set at whose base is a single element; then for

n=(k+1): mk= {mk 1 ) = {{{{m1}}}} we are left with a single

element, a set at whose base is a single element.

Definition 1.2.3: LetUbe the source set and letM, NU. A function,

is a regular function if and only if for all elements m M,

where m is a single element, , where n is a single element.


, is a onto if and only if for all n

N, there exists somem Msuch thaty(m) = n.


, is a one to one if and only if for all nN, there exists a

unique (meaning only one) m Msuch thaty(m) = n.


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Definition 1.2.6: LetUbe the source set and letM, NU. Then we

define a new set, , and this set is

read M cross N.

Theorem 1.2.2: LetUbe a source set and letM, NU. Then the set,MNU.

Proof: Uis the source set, thus it contains all elements ofunsaid

sets, or sets yet to be defined. This means that all elements in set

MNare in U, making MNU.

Definition 1.2.7, 1.2.8: LetUbe a source set and letM, NU. A function,

, where , is defined as an

operation where for all mMand nN,, and Pis the generated setfrom the operation.

Axiom 4: There exists a generated set,

Definition 1.2.9: Given sets M, N, the union ofMand Nis defined as

Definition 1.2.9: A group, M, under operations is defined as

.


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Chapter 2: Building Arithmetic from bottom up

Definition 2.1: A pointis total existence within the smallest possible

distance/area/volume.

Axiom 4: There exist points.

Definition 2.2: Let there be two points,xandy. The distance between

these points is some value, n, where n = |xy|.

Let there be a point, O, the origin, and a point, 1

Let the distance between them be the value called 1.

Plot a point extended in the same direction, the exact same distance

from 1 and call this point 2.

Definition 2.1: Let O be the origin and letn, m be some values

O 1


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Chapter 3: How to count

Did you know that math was created? Thats right Math didnt

just fall from the sky. For every mathematical concept, there was a

being who created it (or discovered it, depending on how you look at it).Lets look at the most basic mathematical skill counting. There are

different ways to count. Each way of counting uses a counting system:

a totally ordered set 2 that can be listed in order starting with a smallest

element.

The most commonly used counting systems in Math are composed

of the Arabic numerals: {0, 1, 2, 3, , 9}. These two sets are called the

whole numbers:W={0, 1, 2, 3, , 8, 9, 10, } and the natural numbers:

N={1, 2, 3, 4, , 9, 10, }. But there are other counting systems that

use different types of numerals. Take for example the Roman numeralcounting system: {I, II, III, IV, V, , X, , L, , C, } or the tally mark

counting system3. These are counting systems that dont use theArabic

numerals. What about the naturalbinary system4 or the natural base 6 5

system? you may ask. Well by golly, these are numbering systems too.

Question 1: There is a big nonreligious book that bases its

content on a counting system. What is this book called? What is

this specific counting system called?

Question 2: Can you create a new counting system? If yes, then

do it. Remember to define a set of numerals first. Define any

possible rules to follow as well.

2 Totally ordered set : A set in which for all elements,xandy, either (1)x>y, (2)

x


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Challenge 1: Can you define the counting system by which [the

book in Question 2]s system is put into order starting with the

least possible element, making use of the dotdotdot?

Challenge 2: Can you use the dotdotdoted listing to define themost general method possible?

book - 1st composition

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