User:Techit/Knowledge dump

Hello, welcome to Techit's knowledge dump page. Here I will dump as many knowledge as possible for collection in the future use.

Mathematics

Mathematics is the study field I hated the most, even though I chose to study in Sci-Math program. Anyways just let's get into the point.

Set

Imagine a data array, it's a set. Set is a collection of different stuff. For example A = {1,2,3,4,5,6,7,8,9} or $\text{Slave}=\{\text{Diskette},\text{GameCreator}\}$.

Elements / Members

A stuff in a data set is called "element" or "member", so to write a set notation we use $\in$(in). So let's see some examples below.

1. $A=\{x|x\in {𝕀}^{+},-5<x\le 12\}$
2. $B=\{x|x\in \mathbb{P},-5\le x\le 23\}$

To translate set notation to normal human-being data set we look at $\in$then use our brain to analyze, after that A should be {x which x in $I^{+}$ (positive integer), x greater than -5 but less or equals to 12}, which in conclusion to be $A=\{1,2,3,4,5,6,7,8,9,10,11,12\}$.

Additional info: you might see these stupid blackboard letters e.g. $ℝ,\mathbb{Q},𝕀,\mathbb{P}$, these can be denotes to...

• $ℝ$ = real numbers
• ${ℝ}^{+}$ = positive real numbers
• ${ℝ}^{-}$ = negative real numbers
• $\mathbb{Q}$ = rational numbers
• ${\mathbb{Q}}^{+}$ = positive real numbers
• ${\mathbb{Q}}^{-}$ = negative real numbers
• $𝕀$ = integer numbers
• ${𝕀}^{+}$ = positive integer numbers
• ${𝕀}^{-}$ = negative integer numbers
• $\mathbb{N}$ = natural numbers
• $\mathbb{P}$ = prime numbers

Venn diagram

Venn diagram is a way to draw set, usually to visualize set operations (usually union, intersection and complement) however it's a pain in ass to draw so I will not talk much about this thing. But for now let's get to know "universe" ($𝕌$; can be written as U, U, 𝓤, Ｕ or whatever the hell you would like) or data range or whatever the heck it's called but in Thai it's called เอกภพสัมพัทธ์.

Subset

Subset is what I hated the most in this topic so let's get fast. So you have $A=\{1,2,3\}$ the A's subset would be $\{\},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}$, if you don't see the pattern, this is the pattern $\{\},\cdots ,A$ where $\cdots$ is combination of every member from 1 member {1} to 2 {1,2} to 3 {1,2,3} members to $\text{n}$ members. The total subset amount will be $n(A)*2$. In this example $n(A)$ is 3 so we will have a total subset of 6.

Here's what you might need to know more about subset;

1. {} is called "empty set" or "null" in computer nerd language
2. $\text{Ø}$ can be used to represent empty set too
3. $A\subseteq B⟺\mathrm{\forall }x(x\in A\Rightarrow x\in B)$ - A is a subset of B if every element of A is also an element of B
4. $A\subset B⟺(A\subseteq B,A\ne B)$ - A is a proper subset only if A is subset of B and A is not equals to B

Power set

Power set is related to subset; it's just a set of all subset in that set. Let's bring $A=\{1,2,3\}$ guess what is $P(A)$? Yes. $P(A)=\{\text{Ø},\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

Basic set operations

Intersection

The intersection of two sets $A$⁠⁠ and $B$ ⁠⁠ is a set denoted ⁠⁠$A\cap B$ whose elements are those elements that belong to both $A$⁠⁠ and ⁠⁠$B$. That is,$A\cap B=\{x|x\in A\wedge x\in B\},$ where $\wedge$ denotes logical and.

--- hi, it's 8:46pm, thats it for today