__Chapter 3__

Counting numbers -> whole numbers -> integers -> rational -> real -> complex

->fractions /

systems of numbers can be similar to what we know as modulo systems

in the book, consider assigning numbers to the days of week, where

Sunday = 0

Monday = 1

Tuesday = 2

Wednesday = 3

Thursday = 4

Friday = 5

Saturday = 6

and then add/subtract 7 from that number, and you can form equations to identify days in a week.

Sunday = 7n

Monday = 7n+1...and so

this allows for higher counting in arithmetic, the development of addition and multiplication tables, and these are further known as congruence classes.

theorem: x^(p-1) = 1 (mod p) for any prime p and any x not congruent to 0 (mod p)

Wilson’s theorem: 1, 2, 3,...(p-1) = -1 (mod p)

__Chapter 4__

sets have developed their own language that represents what is and isn’t part of a set, the book emphasizes the use of {} to represent being in a set. If a number or object is in the brackets, then it is part of a set. Otherwise, it would not be part of the set.

the empty is represented by the cross of a “/” and “0/”

if two sets have the same content, then they are equal

subsets are parts of a whole set

intersections are objects that belong to two sets, and usually there are objects that belong in the individual sets. This is represented by a venn diagram

__Chapter 5__

functions and formulas usually involve x as an independent variable, where x cannot have multiple outputs, but it is possible for multiple inputs to have the same output.

an interesting concept is a function within another functions, because numbers in the first function, must have outputs that are in the domain of the second function

g(f(x))

a function is defined on some set

it takes values in some set

it is defined if a rule is known for finding a value for a given input, and if it’s unique

bijections, or one-to-one correspondence, are exactly those functions which have intervene functions

__Chapter 6__

abstract algebra involves many axioms, which are statements that are known to be true, but not proven to be true

rings, fields, and domains use axioms for all kinds of proofs

an interesting aspect in the set of unions, where intersections add or subtract a sum of combinations

the sum of combinations would be: A + B + C - D - E - F + G

you add the individual section, subtract intersections of two sets, add intersections of three sets, and it continues alternating to account for the intersection of other sets

consider mod 7

[5] + [4] = [2]

[6] x [3] = [4]

congruence is fun

complex numbers involve i, where i^2 = -1

this are the addition and multiplication tables for the congruence class of 4

__Chapter 7__

symmetry is the basic idea that dissecting an object or image or anything and that both sides are the same in some way, whether they are mirror images, or show the same pattern after a rotation, etc.

__Chapter 8__

Euclid’s Axioms:

- any two points lie on a straight line
- two lines meet in at most two points
- any finite line segment may be produced as far as you wish
- it is possible to describe a circle with any center and any radius
- all right angles are equal
- given any line, and any point not on the line, then there exists exactly one line parallel to the first line and passing through the given point

Hilbert gave two requirements for an axiomatic system; completeness, and independence

__Chapter 9__

cardinal traits

- a<=a for any cardinal a
- if a<=B, and B<=y, then a<=y
- if a<=B, and B<=a, then a=B

__Chapter 10__

“Topology is the study of those properties of geometrical objects which remain unchanged under continuous transformations of the object”.

basic objects in topology are topological spaces

if two topological spaces can be passed from one to the other in a continuous way, and back in a continuous way, then it is considered topologically equivalent

__Chapter 11__

Graph theory is the idea behind networks and intersections.

A K_3,3 graph cannot be written/designed without any of the edges not intersecting

A network has two main parts: 1. a set, N, whole elements are called nodes or vertices, 2. a way of specifying when two vertices are joined together

Euler’s Formula: V - E + F = 2

The four color theorem cannot be proven sufficiently, but the five color theorem has been proven to be enough for sections in graph theory. It’s weird since it always seems that four colors is enough to have it so no two areas border each other with the same color

__Chapter 12__

Topological properties: closed curve, connected, disconnected; these prove spheres and toruses are topologically distinct

torus

The sphere, torus, and double torus are orientable

The projective plane, and Klein bottle are not orientable

Topology has a process called “surgery” where spaces were cut up and put back together again

__Chapter 13__

Holes are gaps in topological spaces

Paths are lines that connect two points of a topological space across the space

paths can be used to define functions

__Chapter 14__

1-space (first dimension) = R = the set of real numbers, x

2-space (second) = R^2 = the set of pairs of real numbers (x,y)

3-space (third) = R^3 = the set of triples of real numbers (x,y,z)

since the 4th dimension is not defined, there isn’t any labeling for it yet

the issue comes with “distance”, which is found in the first three dimensions but not in the fourth

distance has three conditions

- the distance between two points is positive
- the distance between two points is the same in either direction
- the distance from A to B should not be linger than A to C plus the distance from C to B

the 5 regular polyhedra in the 3-space are: tetrahedron, cube, octahedron, dodecahedron, icosahedron.

__Chapter 15__

Linear algebraically systems have either one solution, no solutions, or infinitely many solutions

matrices are of the form:

(a b)

(c d)

and vectors look like:

(x)

(y)

and multiplying a matrix by a vector can become another vector or matrix

(a b) (x) = (X) = (ax + by)

(c d) (y) (Y) (cx + dy)

__Chapter 16__

Infinite series are expressions such as

1 + .5 + .25 + .125 + .0625+....

1 + .5 + .25 + .125 + .0625+....+1/2^n = 2 - 1/2^n = 2 - 1/infinity = 2 - 0 = 2

continuous functions do not include jumps, or gaps, in the graph of the function

the line will not break if it is continuous

__Chapter 17__

the chances of throwing a combination of numbers from 2 dice are as follows:

2: 1/36

3: 2/36

4: 3/36

5: 4/36

6: 5/36

7: 6/36

8: 5/36

9: 4/36

10: 3/36

11: 2/36

12: 1/36

This chapter reminds me of using trees in probability, where you can multiple the chances out on the branches to get the end probability for a branch.

Using counting measures such as the choosing, means saying (5 choose 2) is the same as the total number of combinations of 2 items from a set of 5, which in this case is 10.

__Chapter 19__

Using two functions of a profit model can be used to find the optimal combination of each model, and this relates to economics (which is my second major)

__Chapter 20__

If axiomatic set theory is consistent, there exists theorems which can neither be proved nor disproved. There is no constructive procedure which will prove axiomatic set theory to be consistent.

Final Thoughts

This book is awesome, and is a tough read. A lot of the ideas are familiar, and relate back to many studies in high school and college. At the same time, they expand on so many subjects. I definitely feel that I will reread parts of the book, especially the last 3 chapters. At some point, I feel that any mathematics student should read this during their time at a university.

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