WEBVTT

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Game Basics
Game Engine and Game Math 1

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GCC Academy

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Hello everyone

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This is Deukwoo Lee for game maths

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In this lecture

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we will learn about the structure of numbers and function

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First, structure of number is composed of four parts

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Visualization of number, binary operation,

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properties of binary operation

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and field axiom

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I also prepared function and function principles

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I'll start with the basics of function

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and move on to the concept of Cartesian product and calculations related to

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function composition

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Structure of Number

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First let's start with visualization of numbers

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Before that, why do we need to know about numbers?

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In the game world that we'll eventually make

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the basis is a system composed with

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something called vector

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Based on this system, various content is made

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and functions on it

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This vector is in fact

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something made from

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a combination of numbers

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To understand this vector

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we need to understand what system

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and structure numbers have

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When we talk about numbers

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you might have seen this at school

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but there are these many types of numbers

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Number was used along

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the development of human civilization

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when counting, farming, and sailing

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and the concept has advanced

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There are various concepts of numbers

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like natural numbers, integers, rational, irrational

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real, and imaginary numbers

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The concept has expanded and mathematicians

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have used the first letters of each

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from a set perspective

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From here, the set of numbers that we often use

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are real numbers

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or the set of real numbers

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The definition of this

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is the set of numbers no gap between

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a number and another

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Therefore theoretically

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there must be no gap in between numbers

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so there are infinite amounts of numbers in between

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meaning that there are elements

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filling in those spaces

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This concept of the set of real numbers

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and thinking of numbers as elements

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require some practice for you

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and this set

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will be determined depending on

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how we lay out the elements

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If each elements were expressed as

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dots, having no gaps in between

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would make them all be connected

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Therefore the real number set

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can be visualized into this

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straight line that we see in the screen

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We call this the number line

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On this number line

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when we're marking an element

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smaller number goes on the left and the larger on the right

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within the structure of the number line

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this is the way we compare

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big and small of numbers

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But we'll use vectors

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to create systems

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and to become familiar with the vector system

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the way we perceive numbers

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must not be based on just one structure

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but based on 0

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it will be more convenient to

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think of it as two symmetrical structures

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on the right and left

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From this perspective of numbers

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with 0 in the very middle

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a number's property

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can be summarized into two

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First is the distance from the origin

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and second is the structure it belongs in

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between the two

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Those two can be called

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size and direction

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A size

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The size of a number

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is the distance from the origin

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and the direction, so left or right

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there is only two

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So if you're looking for the size of a number

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you can use the absolute value

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to find the distance from the origin

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and for direction

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we can use the + or - sign

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to distinguish them

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Now that we know how the express these numbers

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what we need to look into next is

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about the operation system of numbers

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We often call that binary operation

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We previously talked about

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how numbers are managed in a set concept

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but that concept of sets is actually very simple

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It's just gathering objects we use

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and putting them together into a collection

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Just a collection of elements

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But as you already know

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numbers

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so set of numbers

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is not just a collection of elements called numbers

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but there is also a system called operation within it

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Therefore set of numbers

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are different from any other normal set

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These operations are usually called binary operation

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and as you can see in this picture

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when you put two elements from the number set

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into a machine

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there will be one output

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through the machine

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which is something new

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That's what it is

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From here, we defined an operation

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symbolized by a circle

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and these two elements with the circle operation

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a new product is made

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which is a circle b

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and that again belongs to the set

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then we say that it's closed in math

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These operations, a familiar example would be

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elementary arithmetic

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Addition, subtraction, multiplication, and division

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Let's rethink how we look at

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elementary arithmetic as well

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Let's first think of addition and subtraction

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as a part of the addition and multiplication calculations

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instead of two separate procedures

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We're switching subtraction

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to addition and division

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to multiplication

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As you can see here, we're thinking of

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5-3 as 5+(-3)

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and 5/3 as well into 5*1/3

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a form of multiplication

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Like this let's think of the number system

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with the types of calculations simplified

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from 4 to 2

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And then before getting into the lecture

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we also need to know about how we mark multiplication

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Usually when we write multiplication

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we use the x sign

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But using this x sign

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makes the expression long

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and you might also confuse it with an unknown

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so in math, rather than using the x sign

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it is more usual to use a center dot

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a small dot in the center

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So in this course as well when expressing multiplication

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we won't use x but use a center dot

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for multiplication

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Let's look into addition and multiplication

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I visualized how

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the addition calculation works

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As you can see, 1+(-5)

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It's 1-5 but turned into addition

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What this means is

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to 1, we'll move an element with the size of 5

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to the - direction, that is, left

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parallel

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So 1-5 in addition terms

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is moving the element 1

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by 5 to the left

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2+2=4

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The calculation on the right

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is also the same

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Moving the element 2 to the right by 2

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parallel

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So addition, visually

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is a calculation that moves a dot

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parallel

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Now let's see multiplication

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Multiplication

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is actually addition repeated many times

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But let's rethink it to visualize it

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from a vector perspective

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Multiplication is a calculation on

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how many times we increase

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with the size of the number that we have

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from the origin

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and the concept of direction is added here

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As you can see, 1*4 is

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increasing 1's size four times

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and with the direction that 1 has

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so to the same direction, right, four times

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Then let's see 2*-1

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For this as well, we're taking

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the size of 2 one time, so with

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no change in the scale

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but since there's -1

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we're going

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the opposite direction

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from where we were going

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So it's a 180 degree turn

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So the concept of rotation

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is included in multiplication

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So multiplication of positive number

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is 0 degree rotation

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and multiplication of negative number

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is a 180 degree rotation

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Now let's also take a look at

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the properties of calculations

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also visualized this time as well

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The set of real numbers that we're using

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has the calculation system within it

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So the system included in the

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set with infinite real numbers

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operations, binary operations' properties

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can be divided into three

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Commutativity, associativity, distributivity

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Commutativity is

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that when the operation order is reversed

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the product is the same

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and associativity refers to how

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when there are more than three elements

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the product is the same no matter which operation is done first

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Distributivity

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is the property explained here

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with this expression

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There's something special here

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First there are two types of operations used

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Addition and multiplication

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There's this one and there's this one

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This is left-distributive, and right-distributive

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Both of these must be satisfied

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to be considered to have the property

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of distributivity

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There could be an operation that satisfies

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commutativity, associativity, and distributivity or not

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you just need to remember that these properties exist

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Next thing we need to know is identity element

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It's called identity in English

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Identity is an element

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for a certain element a

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that makes the product of the binary operation

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become the same element a

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That is

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the identity element b

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So in addition

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we input a number

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there is only one element that makes the product a

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It will be 0

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So addition's identity is 0

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and multiplication's identity is 1

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If we expand this concept of identity

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we can obtain the concept of inverse element

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This is a special element that makes the product

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the identity

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Inverse, it might also help

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to know the English

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So when we say that the identity of

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 certain operation is b

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and if we say that a special element c

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always makes a b

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that is the inverse

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Then what is the inverse of addition?

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The production of an operation with a as the input

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must always be 0

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so the inverse of addition is -a

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Like this, putting the opposite sign to a number

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is called the opposite number

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We're going to use this expression later on as well

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so remember that

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and the inverse of multiplication

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since the identity of multiplication is 1

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a fraction with a as the denominator

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makes it 1

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Here, a must not be 0

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which is a special condition

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and taking an element

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as the denominator is called the reciprocal

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We're also going to use this one often

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so remember this

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Addition's inverse is opposite number

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and multiplication's inverse is reciprocal

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And when we mentioned subtraction

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Subtraction is

290
00:16:07.769 --> 00:16:11.700
operation of the opposite number

291
00:16:11.700 --> 00:16:12.859
Same goes for division

292
00:16:12.859 --> 00:16:16.509
Division is the multiplication operation

293
00:16:16.509 --> 00:16:17.940
of the reciprocal

294
00:16:17.940 --> 00:16:21.940
So why do we not use subtraction and division

295
00:16:21.940 --> 00:16:25.140
and only use addition and multiplication?

296
00:16:25.140 --> 00:16:30.050
In subtraction, commutativity does not stand

297
00:16:30.050 --> 00:16:31.100
a-b

298
00:16:31.100 --> 00:16:35.000
For example, 3-5

299
00:16:35.000 --> 00:16:37.099
in an subtraction expression

300
00:16:37.099 --> 00:16:39.859
the product is not the same as 5-3

301
00:16:39.859 --> 00:16:44.659
Reversing a subtraction always gives the opposite number

302
00:16:44.659 --> 00:16:49.059
which means that it does not satisfy commutativity

303
00:16:49.059 --> 00:16:51.559
But with addition

304
00:16:51.559 --> 00:16:56.420
if we change 3-5 to 3+(-5)

305
00:16:56.420 --> 00:17:00.420
it gives the same product as (-5)+3

306
00:17:00.420 --> 00:17:02.659
so commutativity is satisfied

307
00:17:02.659 --> 00:17:04.540
Same for multiplication

308
00:17:04.540 --> 00:17:09.780
Therefore we don't use subtraction and division

309
00:17:09.780 --> 00:17:11.477
but use only addition

310
00:17:11.477 --> 00:17:16.380
and multiplication to express all operations

311
00:17:16.380 --> 00:17:19.780
Then using only these two operations

312
00:17:19.780 --> 00:17:23.030
let's analyse the operation structure

313
00:17:23.030 --> 00:17:24.515
of real numbers

314
00:17:27.614 --> 00:17:31.780
When we analyse the structure a number has

315
00:17:31.780 --> 00:17:36.900
we use axiom

316
00:17:36.900 --> 00:17:39.600
This axiom means the most basic proposition

317
00:17:39.600 --> 00:17:44.339
that does not need a proof in a theory structure

318
00:17:44.339 --> 00:17:47.500
So we'll analyse the axioms

319
00:17:47.500 --> 00:17:51.739
and also the set of numbers that satisfy the axioms

320
00:17:51.739 --> 00:17:52.989
to create

321
00:17:52.989 --> 00:17:56.140
a structure of numbers that

322
00:17:56.140 --> 00:17:58.640
satisfy the properties

323
00:17:58.640 --> 00:18:01.339
instead of what we actually use

324
00:18:01.339 --> 00:18:04.087
First there is axiom of group

325
00:18:06.651 --> 00:18:09.001
If there are two operations

326
00:18:09.001 --> 00:18:11.420
on a certain number structure

327
00:18:11.420 --> 00:18:13.820
and if there is a number structure

328
00:18:13.820 --> 00:18:17.420
that satisfies the following axiom

329
00:18:17.420 --> 00:18:21.459
then we say that it has group structure

330
00:18:21.459 --> 00:18:24.939
Let's try addition first

331
00:18:24.939 --> 00:18:26.089
The addition

332
00:18:26.089 --> 00:18:29.459
must be closed

333
00:18:29.459 --> 00:18:31.709
meaning that the product

334
00:18:31.709 --> 00:18:34.939
of the addition must be in the set of numbers used

335
00:18:34.939 --> 00:18:39.099
It must be associative and an identity must exist

336
00:18:39.099 --> 00:18:42.754
The identity of real number is 0

337
00:18:42.764 --> 00:18:44.890
so we can see that this is immediately checked

338
00:18:44.890 --> 00:18:47.459
and inverse as well

339
00:18:47.459 --> 00:18:52.020
There is an opposite number so the inverse of addition exists

340
00:18:52.020 --> 00:18:56.720
Therefore real numbers

341
00:18:56.720 --> 00:18:58.579
have group structure

342
00:18:58.579 --> 00:19:02.739
We use parentheses and a comma

343
00:19:02.739 --> 00:19:07.660
and R, the real number symbol and a + sign

344
00:19:07.660 --> 00:19:11.099
to say that it has group structure

345
00:19:11.099 --> 00:19:14.780
Another condition comes

346
00:19:14.780 --> 00:19:16.980
If it is commutative

347
00:19:16.980 --> 00:19:20.339
it can also be said that it is in the Abelian group

348
00:19:20.339 --> 00:19:24.260
Like this, axioms accumulate

349
00:19:24.260 --> 00:19:27.510
and what comes next is another axiom

350
00:19:27.510 --> 00:19:30.099
for the second operation

351
00:19:30.099 --> 00:19:31.149
Multiplication

352
00:19:31.149 --> 00:19:34.420
Let's see what there is for multiplication

353
00:19:34.420 --> 00:19:37.819
The product of multiplication is always a real number

354
00:19:37.819 --> 00:19:42.579
therefore it's closed

355
00:19:42.579 --> 00:19:44.879
So axiom of ring

356
00:19:44.879 --> 00:19:48.380
means that it satisfies that it's closed

357
00:19:48.380 --> 00:19:50.280
for the second operation

358
00:19:50.280 --> 00:19:52.579
and associativity and distributivity

359
00:19:52.579 --> 00:19:55.339
Distributivity uses two operations

360
00:19:55.339 --> 00:19:58.535
so the first operation in group

361
00:19:58.535 --> 00:20:01.535
and the second operation in ring are combined

362
00:20:01.535 --> 00:20:03.435
and if they satisfy distributivity

363
00:20:03.435 --> 00:20:06.800
it means that this has ring structure

364
00:20:06.800 --> 00:20:11.350
We can also simplicize this

365
00:20:11.350 --> 00:20:15.440
with signs

366
00:20:15.440 --> 00:20:18.187
Then we're adding another

367
00:20:18.187 --> 00:20:21.640
more strict condition here

368
00:20:21.640 --> 00:20:25.559
For the second operation, it must be distributive

369
00:20:25.559 --> 00:20:29.331
and have identity

370
00:20:29.331 --> 00:20:32.599
then we say that it has the commutative ring structure

371
00:20:32.599 --> 00:20:35.549
Lastly, multiplication

372
00:20:35.549 --> 00:20:37.883
If there is inverse

373
00:20:37.883 --> 00:20:40.000
for the second operation

374
00:20:40.000 --> 00:20:42.750
we say that it has field structure

375
00:20:42.750 --> 00:20:45.000
So except for 0

376
00:20:45.000 --> 00:20:48.200
since an inverse cannot be 0

377
00:20:48.200 --> 00:20:51.500
So for all operations except for 0

378
00:20:51.500 --> 00:20:53.749
if there is inverse for the second operation

379
00:20:53.749 --> 00:20:57.759
then it has field structure

380
00:20:57.759 --> 00:21:00.659
So real numbers we normally use

381
00:21:00.659 --> 00:21:02.480
have field structure

382
00:21:02.480 --> 00:21:07.559
because it satisfies all axioms mentioned

383
00:21:07.559 --> 00:21:10.409
So to gather up everything we've talked about

384
00:21:10.409 --> 00:21:13.559
if it's closed for addition and multiplication

385
00:21:13.559 --> 00:21:16.909
satisfies commutativity, associativity, distributivity

386
00:21:16.909 --> 00:21:20.680
and has identity and inverse

387
00:21:20.680 --> 00:21:22.780
so to put it simply

388
00:21:22.780 --> 00:21:27.839
it must be closed for elementary arithmetic and the operation can freely be expanded

389
00:21:27.839 --> 00:21:31.959
The operation order is flexible

390
00:21:31.959 --> 00:21:35.059
The set of numbers satisfying this field structure

391
00:21:35.059 --> 00:21:39.120
include real number and also rational number and complex number

392
00:21:39.120 --> 00:21:41.846
So when dealing with numbers

393
00:21:41.846 --> 00:21:44.959
although we usually are usually in the real number structure

394
00:21:44.959 --> 00:21:50.059
theoretically, rather than limiting it to real number

395
00:21:50.059 --> 00:21:54.328
it is better to say that we use the set of numbers

396
00:21:54.328 --> 00:21:58.228
that have field structure

397
00:21:58.228 --> 00:22:01.738
to expand the theory

398
00:22:01.738 --> 00:22:03.988
And based on that

399
00:22:03.988 --> 00:22:07.199
various systems are expanded

400
00:22:07.199 --> 00:22:10.499
 This kind of set with field structure

401
00:22:10.499 --> 00:22:12.880
are expressed as F

402
00:22:12.880 --> 00:22:16.280
and each element of that set

403
00:22:16.280 --> 00:22:20.720
is called scalar

404
00:22:20.720 --> 00:22:24.217
Now we compared the sets

405
00:22:24.217 --> 00:22:28.720
that we use and the theoretical one

406
00:22:28.720 --> 00:22:31.370
For real numbers

407
00:22:31.370 --> 00:22:34.439
I mentioned that we use the symbol R

408
00:22:34.439 --> 00:22:37.399
and the elements are real numbers

409
00:22:37.399 --> 00:22:39.849
But in the concept of field set

410
00:22:39.849 --> 00:22:42.240
from a structural concept

411
00:22:42.240 --> 00:22:44.590
it is called the field set F

412
00:22:44.590 --> 00:22:49.199
and the elements are scalars

413
00:22:49.199 --> 00:22:51.907
Therefore when we see

414
00:22:51.907 --> 00:22:55.407
the operation a+b

415
00:22:55.407 --> 00:22:59.639
it is the addition of real numbers a and b

416
00:22:59.639 --> 00:23:01.089
and this

417
00:23:01.089 --> 00:23:03.908
has a limit in a way since although we are

418
00:23:03.908 --> 00:23:09.559
flexible with the order of operations, the product is always real numbers

419
00:23:09.559 --> 00:23:11.459
But that expression

420
00:23:11.459 --> 00:23:15.360
expressed as the addition of scalar a and scalar b

421
00:23:15.360 --> 00:23:19.458
this means that all sets that fulfill the property of field

422
00:23:19.458 --> 00:23:22.759
No matter rational number, complex number, or real number

423
00:23:22.759 --> 00:23:25.717
satisfying commutativity, associativity, and distributivity

424
00:23:25.717 --> 00:23:29.117
and having identity and inverse becomes the

425
00:23:29.117 --> 00:23:32.760
satisfying condition for the number set that we can

426
00:23:32.760 --> 00:23:35.600
use more freely and commonly

427
00:23:35.600 --> 00:23:40.039
Based on this field set of numbers

428
00:23:40.039 --> 00:23:44.960
we can expand to what we're going to learn next

429
00:23:44.960 --> 00:23:46.365
vector

430
00:23:46.997 --> 00:23:50.607
Function

431
00:23:50.940 --> 00:23:54.798
Before learning the basics of function

432
00:23:54.798 --> 00:23:58.881
let's see why we need to learn function first

433
00:23:58.881 --> 00:24:01.523
The game we're going to make is at the end of the day

434
00:24:01.523 --> 00:24:08.000
very closely related to the transformation of the game space

435
00:24:08.000 --> 00:24:13.000
Creating a game is closely tied to numerous

436
00:24:13.000 --> 00:24:17.000
space transformation processes

437
00:24:17.000 --> 00:24:20.850
and the space transformation here

438
00:24:20.850 --> 00:24:25.000
ultimately stems from

439
00:24:25.000 --> 00:24:27.400
sets

440
00:24:27.400 --> 00:24:29.792
and correspondence of sets

441
00:24:29.792 --> 00:24:34.624
What explains the mechanism of this is function

442
00:24:34.624 --> 00:24:37.024
so wee must know function well

443
00:24:37.024 --> 00:24:39.851
and understand the concept well

444
00:24:39.851 --> 00:24:42.901
so that we can use that as the basis to

445
00:24:42.901 --> 00:24:47.495
understanding the space and transformation of spaces

446
00:24:47.495 --> 00:24:51.307
Then let's get into function

447
00:24:51.307 --> 00:24:54.852
The definition of function can be

448
00:24:54.852 --> 00:24:57.802
When there are two sets, X and Y

449
00:24:57.802 --> 00:25:01.832
a correspondence between the two sets

450
00:25:01.832 --> 00:25:07.232
So element x from the first set X

451
00:25:07.232 --> 00:25:12.119
corresponds to element y of the second set

452
00:25:12.119 --> 00:25:16.465
then that is expressed as y=f(x)

453
00:25:16.465 --> 00:25:17.865
This function

454
00:25:17.865 --> 00:25:21.455
isn't just made from the correspondence only

455
00:25:21.455 --> 00:25:25.000
there are more

456
00:25:25.000 --> 00:25:28.850
First is that all elements of the first set

457
00:25:28.850 --> 00:25:31.000
must be used

458
00:25:33.000 --> 00:25:36.000
Let me try showing you an example of

459
00:25:36.000 --> 00:25:37.643
correspondence that is not a function

460
00:25:37.643 --> 00:25:39.093
As I mentioned

461
00:25:39.093 --> 00:25:41.762
all elements of the first set must be used

462
00:25:41.762 --> 00:25:46.012
That means an element of the first set

463
00:25:46.012 --> 00:25:48.495
having no correspondence

464
00:25:48.495 --> 00:25:51.663
makes that impossible to be considered a function

465
00:25:51.663 --> 00:25:55.291
Next, when an element of the first set

466
00:25:55.291 --> 00:25:57.733
corresponds to two elements of the second set

467
00:25:57.733 --> 00:26:01.000
we also say that that doesn't satisfy the conditions of being a function

468
00:26:01.000 --> 00:26:04.574
Let me explain using this diagram

469
00:26:04.574 --> 00:26:10.074
Here, elements 3 and 4 of the first set X

470
00:26:10.074 --> 00:26:12.000
has no correspondence

471
00:26:12.000 --> 00:26:15.178
so it does not satisfy the conditions of a function

472
00:26:15.178 --> 00:26:19.278
And the element 2 of the first set

473
00:26:19.278 --> 00:26:22.028
corresponds to two elements B and C

474
00:26:22.028 --> 00:26:23.485
of the second set

475
00:26:23.485 --> 00:26:26.835
Therefore this also does not satisfy

476
00:26:26.835 --> 00:26:30.406
the condition of being a function

477
00:26:30.406 --> 00:26:33.406
Here, in this correspondence

478
00:26:33.406 --> 00:26:36.000
what we need to know are function vocabularies

479
00:26:36.000 --> 00:26:38.250
These vocabs are

480
00:26:38.250 --> 00:26:41.812
Since function is a very basic field study

481
00:26:41.812 --> 00:26:43.712
it is used in

482
00:26:43.712 --> 00:26:45.662
very many places

483
00:26:45.662 --> 00:26:49.139
so let's get into the vocabularies

484
00:26:49.139 --> 00:26:52.589
We call the first set

485
00:26:52.589 --> 00:26:55.624
the domain

486
00:26:55.624 --> 00:26:58.374
and the corresponding second set

487
00:26:58.374 --> 00:27:01.594
the codomain

488
00:27:01.594 --> 00:27:05.344
Although all elements of the domain must correspond

489
00:27:05.344 --> 00:27:08.844
the elements of the second set

490
00:27:08.844 --> 00:27:10.802
need not

491
00:27:10.802 --> 00:27:14.352
Therefore there will be cases where

492
00:27:14.352 --> 00:27:17.039
only parts of the second set are corresponded, so

493
00:27:17.039 --> 00:27:21.139
we make another set with only the corresponded elements

494
00:27:21.139 --> 00:27:25.475
and that is called the range

495
00:27:25.475 --> 00:27:28.275
That was the concept of domain

496
00:27:28.275 --> 00:27:29.723
codomain, and range

497
00:27:29.723 --> 00:27:33.773
This function, from a programming or mechanical point of view

498
00:27:33.773 --> 00:27:36.851
similar to binary operation from lecture 1

499
00:27:36.851 --> 00:27:39.451
is a system that gives

500
00:27:39.451 --> 00:27:41.631
a new element

501
00:27:41.631 --> 00:27:44.297
when we input one

502
00:27:44.297 --> 00:27:49.440
So the input is

503
00:27:49.440 --> 00:27:52.327
an element of the domain

504
00:27:52.327 --> 00:27:54.927
and the output is

505
00:27:54.927 --> 00:27:57.366
the corresponded element of the codomain

506
00:27:57.366 --> 00:28:00.315
We can divide functions into group

507
00:28:00.315 --> 00:28:02.584
according to the correspondence type

508
00:28:02.584 --> 00:28:05.684
As you can see here, if the codomain

509
00:28:05.684 --> 00:28:08.475
and the range are the same

510
00:28:08.475 --> 00:28:11.772
So all elements of the codomain are corresponded

511
00:28:11.772 --> 00:28:13.761
This case is called

512
00:28:13.761 --> 00:28:16.000
surjection

513
00:28:16.000 --> 00:28:18.650
Next is when elements of the domain and the codomain

514
00:28:18.650 --> 00:28:21.646
are corresponded 1:1

515
00:28:21.646 --> 00:28:24.039
which is injection

516
00:28:24.039 --> 00:28:27.439
The surjection that I mentioned previously

517
00:28:27.439 --> 00:28:29.539
has the element c which is corresponded

518
00:28:29.539 --> 00:28:32.455
to two elements of the domain, 3, and 4

519
00:28:32.455 --> 00:28:35.944
so this is not a 1:1 correspondence

520
00:28:35.944 --> 00:28:39.713
Therefore this does not have the property of an injection

521
00:28:39.713 --> 00:28:42.460
On the other hand for the injection

522
00:28:42.460 --> 00:28:45.851
element c in the codomain is not corresponded

523
00:28:45.851 --> 00:28:47.951
Therefore the codomain and the range

524
00:28:47.951 --> 00:28:49.970
are different sets

525
00:28:49.970 --> 00:28:52.170
Therefore this does not satisfy

526
00:28:52.170 --> 00:28:55.386
the conditions of a surjection

527
00:28:55.386 --> 00:28:59.584
Then what happens when it satisfies both surjection and injection?

528
00:28:59.584 --> 00:29:02.634
Then all elements of

529
00:29:02.634 --> 00:29:06.807
the domain and all elements of the codomain

530
00:29:06.807 --> 00:29:10.742
are corresponded 1:1

531
00:29:10.742 --> 00:29:14.515
This correspondence is called bijection

532
00:29:14.515 --> 00:29:18.365
This satisfies both properties of

533
00:29:18.365 --> 00:29:21.911
surjection and injection

534
00:29:21.911 --> 00:29:25.089
Then is surjection, injection, and bijection all?

535
00:29:25.089 --> 00:29:30.089
There are also correspondences that are neither of these three

536
00:29:30.089 --> 00:29:34.406
You can see here this is not a surjection

537
00:29:34.406 --> 00:29:36.156
because the codomain and the range is different

538
00:29:36.156 --> 00:29:38.772
so it does not satisfy the condition of a surjection

539
00:29:38.772 --> 00:29:40.222
and for element c

540
00:29:40.222 --> 00:29:43.277
elements 3 and 4 are corresponded

541
00:29:43.277 --> 00:29:46.627
so it also does not satisfy the condition of injection

542
00:29:46.627 --> 00:29:49.267
which makes it just a normal function

543
00:29:51.040 --> 00:29:54.297
Next let's see Cartesian product

544
00:29:54.297 --> 00:29:55.947
Cartesian product

545
00:29:55.947 --> 00:30:00.515
is picking elements from two different sets

546
00:30:00.515 --> 00:30:04.544
and putting them together into an ordered pair

547
00:30:04.544 --> 00:30:11.320
So it's a new set

548
00:30:11.320 --> 00:30:13.733
made with elements from two sets

549
00:30:13.733 --> 00:30:16.083
So like (a,b)

550
00:30:16.083 --> 00:30:18.483
using parentheses and comma to express this

551
00:30:18.483 --> 00:30:20.465
is called a tuple

552
00:30:20.465 --> 00:30:25.436
Here, a is an element from set A

553
00:30:25.436 --> 00:30:26.627
and b over here

554
00:30:26.627 --> 00:30:29.772
is an element from set B

555
00:30:29.772 --> 00:30:32.322
This is a new set

556
00:30:32.322 --> 00:30:34.574
combining two sets

557
00:30:34.574 --> 00:30:37.924
This Cartesian product

558
00:30:37.924 --> 00:30:40.624
uses the multiplication sign, x

559
00:30:40.624 --> 00:30:44.554
to the used sets

560
00:30:44.554 --> 00:30:47.079
Let's see an example

561
00:30:47.079 --> 00:30:50.544
Set a has two elements a and b

562
00:30:50.544 --> 00:30:54.901
and set B has two elements 1 and 2

563
00:30:54.901 --> 00:30:58.001
Using these sets to make a Cartesian product

564
00:30:58.001 --> 00:31:01.535
will correspond 1 and 2 to a

565
00:31:01.535 --> 00:31:04.635
so there will be new elements

566
00:31:04.635 --> 00:31:07.297
(a,1) and (a,2)

567
00:31:07.297 --> 00:31:10.197
For the second element b of set A

568
00:31:10.197 --> 00:31:13.287
1 and 2 is corresponded again

569
00:31:13.287 --> 00:31:17.178
which makes (b,1) and (b,2)

570
00:31:17.178 --> 00:31:20.678
So the set composed of four elements in total

571
00:31:20.678 --> 00:31:22.584
is made

572
00:31:22.584 --> 00:31:24.822
and this is called the Cartesian product

573
00:31:24.822 --> 00:31:26.272
Using the two sets

574
00:31:26.272 --> 00:31:30.416
we write it as AxB

575
00:31:30.416 --> 00:31:33.066
A good example of Cartesian product is

576
00:31:33.066 --> 00:31:35.178
I brought this from Wikipedia

577
00:31:35.178 --> 00:31:38.574
You can think of the Trump cards

578
00:31:38.574 --> 00:31:43.596
The number on the Trump cards

579
00:31:43.596 --> 00:31:46.059
can be one set

580
00:31:46.059 --> 00:31:48.861
and the sign another set

581
00:31:48.861 --> 00:31:51.711
These two sets R and S

582
00:31:51.711 --> 00:31:54.317
made into a Cartesian product

583
00:31:54.317 --> 00:31:56.899
which makes the Trump cards

584
00:31:56.899 --> 00:31:59.861
composed of 52 cards

585
00:31:59.861 --> 00:32:03.511
This Cartesian product concept

586
00:32:03.511 --> 00:32:07.297
is used later on when we learn coordinate system

587
00:32:07.297 --> 00:32:10.832
but it can be used in the binary operation concept as well

588
00:32:10.832 --> 00:32:13.132
When we were learning function

589
00:32:13.132 --> 00:32:16.218
there was only one input

590
00:32:16.218 --> 00:32:19.148
but in binary operation

591
00:32:19.148 --> 00:32:21.448
that we learned in the previous lecture

592
00:32:21.448 --> 00:32:23.673
there were two inputs

593
00:32:23.673 --> 00:32:26.922
So you might think that binary operation can be hard

594
00:32:26.922 --> 00:32:29.614
to express in function forms

595
00:32:29.614 --> 00:32:35.693
but if we use Cartesian products

596
00:32:35.693 --> 00:32:38.673
on the domains

597
00:32:38.673 --> 00:32:43.148
we can put two elements as inputs

598
00:32:43.148 --> 00:32:46.931
so making one product with two inputs

599
00:32:46.931 --> 00:32:50.723
is possible from binary operation to function as well

600
00:32:50.723 --> 00:32:56.623
So for operations like a+b

601
00:32:56.623 --> 00:32:59.327
done in real number set

602
00:32:59.327 --> 00:33:05.030
the domain is made with a Cartesian product of real number set

603
00:33:05.030 --> 00:33:08.080
R x R, like this

604
00:33:08.080 --> 00:33:10.139
we make a ordered pair of two

605
00:33:10.139 --> 00:33:13.812
and put each elements as inputs

606
00:33:13.812 --> 00:33:15.670
to create the output

607
00:33:15.670 --> 00:33:19.000
and we have a perfect system that does binary operation

608
00:33:19.000 --> 00:33:22.158
as a function

609
00:33:22.158 --> 00:33:25.624
Next is composition function

610
00:33:25.624 --> 00:33:30.515
Composition is when there are two functions

611
00:33:30.515 --> 00:33:33.815
and three sets

612
00:33:33.815 --> 00:33:38.215
Here, from set X to set Z

613
00:33:38.215 --> 00:33:41.495
exists a correspondence

614
00:33:41.495 --> 00:33:43.006
In the case of element 4

615
00:33:43.006 --> 00:33:47.198
it first corresponds to element D of the second set

616
00:33:47.198 --> 00:33:49.485
through the first function f(x)

617
00:33:49.485 --> 00:33:53.085
and then through g(y) it corresponds to

618
00:33:53.085 --> 00:33:55.742
alpha of the third set

619
00:33:55.742 --> 00:33:59.881
It proceeds step by step

620
00:33:59.881 --> 00:34:04.723
If we get rid of the middle part

621
00:34:04.723 --> 00:34:08.412
all elements 4, 3, and 2

622
00:34:08.412 --> 00:34:12.238
correspond to alpha

623
00:34:12.238 --> 00:34:14.788
So getting rid of the middle part

624
00:34:14.788 --> 00:34:17.000
and going straight to

625
00:34:17.000 --> 00:34:19.500
the last step

626
00:34:19.500 --> 00:34:22.000
is what composition is

627
00:34:22.000 --> 00:34:25.000
and this means that

628
00:34:25.000 --> 00:34:28.000
two functions were composed

629
00:34:28.000 --> 00:34:30.198
and the order is important here

630
00:34:30.198 --> 00:34:33.198
First function f

631
00:34:33.198 --> 00:34:36.000
and the second function g

632
00:34:36.000 --> 00:34:40.000
but we write the first function on the right

633
00:34:40.000 --> 00:34:45.287
so it becomes g ∘ f

634
00:34:45.287 --> 00:34:48.787
Using parentheses

635
00:34:48.787 --> 00:34:52.594
it performs f(x) first and then g(x)

636
00:34:52.594 --> 00:34:57.495
so it's correct to have g on the left

637
00:34:57.495 --> 00:34:59.745
but when we use this sign

638
00:34:59.745 --> 00:35:02.693
you might confuse that g is performed first

639
00:35:02.693 --> 00:35:06.277
But what's on the right is what's performed first

640
00:35:06.277 --> 00:35:10.653
I hope you remember this so you don't get confused

641
00:35:10.653 --> 00:35:13.297
Before getting into the properties of composite functions

642
00:35:13.297 --> 00:35:17.228
let's first learn about some correspondence types

643
00:35:17.228 --> 00:35:18.828
Codomain and domain

644
00:35:18.828 --> 00:35:24.178
In bijection where domain and codomain are corresponded

645
00:35:24.178 --> 00:35:29.109
So the elements of the domain, codomain, and range are the same

646
00:35:29.109 --> 00:35:34.812
This correspondence is called

647
00:35:34.812 --> 00:35:37.212
identity function

648
00:35:37.212 --> 00:35:38.861
and uses i as the symbol

649
00:35:38.861 --> 00:35:41.713
But identity was also mentioned before

650
00:35:41.713 --> 00:35:46.891
It's the same, identity function

651
00:35:46.891 --> 00:35:49.614
And we had also learned about the inverse

652
00:35:49.614 --> 00:35:52.079
There is also a similar concept

653
00:35:52.079 --> 00:35:54.198
in function

654
00:35:54.198 --> 00:35:56.059
and this is called

655
00:35:56.059 --> 00:35:57.909
inverse function

656
00:35:57.909 --> 00:36:02.366
It is the same concept as inverse

657
00:36:02.366 --> 00:36:05.881
This holds for bijection

658
00:36:05.881 --> 00:36:11.782
So conversing a correspondence from the domain to the codomain

659
00:36:11.782 --> 00:36:16.132
to one from the codomain to the domain

660
00:36:16.132 --> 00:36:19.356
is the inverse function

661
00:36:19.356 --> 00:36:23.564
This has some conditions

662
00:36:23.564 --> 00:36:25.079
so let's take a look

663
00:36:25.079 --> 00:36:28.841
First is when it's a surjection

664
00:36:28.841 --> 00:36:35.091
Let's say that A,B, and C in the codomain

665
00:36:35.091 --> 00:36:38.000
are conversely being corresponded to the domain

666
00:36:38.000 --> 00:36:43.129
Then A is corresponded to 1 and B to 2

667
00:36:43.129 --> 00:36:47.673
but C is corresponded to both 3 and 4

668
00:36:47.673 --> 00:36:51.623
In this case, this does not satisfy the condition of a function

669
00:36:51.623 --> 00:36:56.257
so an inverse function does not exist

670
00:36:56.257 --> 00:36:58.643
Let's see the case of an injection

671
00:36:58.643 --> 00:37:03.594
Here, A is corresponded to 1 and B to 2

672
00:37:03.594 --> 00:37:07.386
but C doesn't have a correspondence

673
00:37:07.386 --> 00:37:10.986
This again does not satisfy the condition of a function

674
00:37:10.986 --> 00:37:15.148
so injection also does not have an inverse function

675
00:37:15.148 --> 00:37:17.822
It does not exist

676
00:37:17.822 --> 00:37:21.572
Lastly when it's a bijection

677
00:37:21.572 --> 00:37:24.000
all elements are corresponded 1:1

678
00:37:24.000 --> 00:37:26.634
and without anything left

679
00:37:26.634 --> 00:37:29.634
so a bijection always

680
00:37:29.634 --> 00:37:33.000
has its inverse function

681
00:37:33.000 --> 00:37:35.700
So to make an inverse function

682
00:37:35.700 --> 00:37:39.250
the function must be a

683
00:37:39.250 --> 00:37:41.000
bijection

684
00:37:41.000 --> 00:37:44.517
Remembering this condition

685
00:37:44.517 --> 00:37:49.000
will help so I'll keep mentioning this

686
00:37:49.000 --> 00:37:55.000
And when there is a function and its inverse function

687
00:37:55.000 --> 00:37:59.000
this function must be a bijection as mentioned just before

688
00:37:59.000 --> 00:38:00.850
When there is a bijection

689
00:38:00.850 --> 00:38:03.366
what happens when we compose the inverse function?

690
00:38:03.366 --> 00:38:06.124
This is the correspondence

691
00:38:06.124 --> 00:38:07.724
that went to the codomain

692
00:38:07.724 --> 00:38:10.832
coming back to the domain again

693
00:38:10.832 --> 00:38:14.574
so it comes back to itself

694
00:38:14.574 --> 00:38:16.974
therefor it has the same result

695
00:38:16.974 --> 00:38:20.436
as the previously mentioned identity function

696
00:38:20.436 --> 00:38:25.586
Therefore the composition of a bijection and its inverse function

697
00:38:25.586 --> 00:38:30.168
always becomes an identity function

698
00:38:30.168 --> 00:38:32.168
So this in symbols is

699
00:38:32.168 --> 00:38:40.030
f ∘ f^-1 = i

700
00:38:42.614 --> 00:38:45.065
Then let's see the inverse function

701
00:38:45.065 --> 00:38:47.733
of composite function

702
00:38:47.733 --> 00:38:53.158
Let's say we have two bijections

703
00:38:53.158 --> 00:38:57.408
We said that performing f first and then g

704
00:38:57.408 --> 00:38:59.416
was written as g ∘ f

705
00:38:59.416 --> 00:39:02.066
The inverse of this composite function

706
00:39:02.066 --> 00:39:04.079
as you can see in the diagram

707
00:39:04.079 --> 00:39:08.653
is performing the inverse of g first

708
00:39:08.653 --> 00:39:11.303
and then performing

709
00:39:11.303 --> 00:39:13.253
the inverse of f

710
00:39:13.253 --> 00:39:16.535
which is obvious

711
00:39:16.535 --> 00:39:22.178
Therefore the inverse of g ∘ f

712
00:39:22.178 --> 00:39:24.663
What's performed first is g^-1

713
00:39:24.663 --> 00:39:27.148
so it must come on the right

714
00:39:27.148 --> 00:39:30.089
and what's done second is f^-1

715
00:39:30.089 --> 00:39:33.596
so it becomes

716
00:39:33.596 --> 00:39:36.148
f^-1 ∘ g^-1

717
00:39:36.148 --> 00:39:40.010
Why do we need to know the correspondence of composite functions?

718
00:39:40.010 --> 00:39:45.525
Later when we deal with space transformations

719
00:39:45.525 --> 00:39:48.396
we will use matrix

720
00:39:48.396 --> 00:39:52.584
and matrix multiplication is

721
00:39:52.584 --> 00:39:55.871
basically composite function

722
00:39:55.871 --> 00:39:58.771
So understanding the mechanism

723
00:39:58.771 --> 00:40:01.445
of composite function

724
00:40:01.445 --> 00:40:05.703
becomes the basis for understanding

725
00:40:05.703 --> 00:40:07.000
the complex mechanism of matrix multiplication

726
00:40:07.000 --> 00:40:11.350
Therefore from basic knowledge

727
00:40:11.350 --> 00:40:14.356
from sets and correspondence of sets

728
00:40:14.356 --> 00:40:19.148
then composing them and how inverse works

729
00:40:19.158 --> 00:40:22.307
learning these during this lecture

730
00:40:22.307 --> 00:40:25.990
was for that reason

731
00:40:25.990 --> 00:40:28.178
This is it for this lecture

732
00:40:28.178 --> 00:40:30.208
Good job

733
00:40:30.208 --> 00:40:31.208
Thank you

734
00:40:31.524 --> 00:40:32.516
Structure of number
Visualization of numbers
Vectors, combinations of numbers compose game content system therefore understanding numbers is needed

735
00:40:32.516 --> 00:40:33.539
Types of numbers: natural, integers, rational, irrational, real, complex numbers
Must get used to the systems based on the origin to left and right

736
00:40:33.539 --> 00:40:34.547
Binary operation
Reconfiguration by replacing subtraction with addition and division with multiplication of the  elementary arithmetics

737
00:40:34.547 --> 00:40:35.541
Visualization of Addition: move points in parallel
Visualization of multiplication: adjusting the fold and direction of increase from the origin

738
00:40:35.541 --> 00:40:36.543
Properties of binary operations
Commutativity, associativity, distributivity
Identity: makes a's product a
Inverse: makes a's product the identity

739
00:40:36.543 --> 00:40:39.043
Field axiom
Axiom: the most basic proposition that doesn not need proof
Used axiom to analyse and build number structures

740
00:40:39.043 --> 00:40:41.523
Real, rational, complex numbers satisfy field structure
All numbers satisfying the field structure can be used

741
00:40:41.523 --> 00:40:43.122
Function
Basics of function
Only when the concept of function is established can the transformation of space be understood correctly

742
00:40:43.122 --> 00:40:44.827
Function: Correspondence of two sets
Domain: First set
Codomain: the corresponding second set
Range: a set of corresponding elements in the codomain

743
00:40:44.827 --> 00:40:46.532
Surjection: all elements of codomain are in the range
Injection: domain and codomain correspond 1:1
Bijection: all elements correspond 1:1

744
00:40:46.532 --> 00:40:48.214
Cartesian product
Set of ordered pairs of elements from both two sets
Using this as the domain allows binary operation to be expressed as functions

745
00:40:48.214 --> 00:40:49.933
Composition
Identity function: all elements correspond to itself
Inverse function: inverse correspondence
Bijection condition for inverse function

746
00:40:49.933 --> 00:40:51.644
Transformations of space use matrices and matrix multiplication correspond to composite functions, making such knowledge necessary