date: 2024-10-04
title: 02-TOC-Math
status: DONE
author:
- AllenYGY
tags:
- NOTE
- TOC
- Math
publish: True
TOC-Math
An object
To represent a set, you need to
"
A universal set is a set that contains all elements under a certain context.
A universal set is usually denoted in the blackboard bold font. For example,
Set
Let A be a set.
Let
Suppose
A complement is the set
A intersection
A union
A setminus
Suppose
The cartesian product of
Each object
A relation
If
A relation
Let
A relation
If
To denote a function:
For the same example, one can also write
Please distinguish “
For example, let
To precisely describe mathematics, algorithm, computation, and other procedures, formal languages are used. A language is a set of strings over an alphabet. Formally,
A language is a set of strings over a specific alphabet.
An alphabet is a nonempty finite set.
For example,
A string over an alphabet
If
The empty string is the string of length 0, denoted by
Let
The reverse of a string
The string
The concatenation of two strings
For example,
Sometimes, the operator “
A language is a set of strings over a specific alphabet.
In this lecture, we only emphasize several proving techniques.
The cardinality of a set
The set
Note that the cardinality can be finite and infinite.
A set is countable if it is finite or its cardinality is same as the set of natural number
Countable means that:
In fact, the counting procedure constructs a bijection, mapping a natural number
We denote the cardinality of
To proof an infinite set
Georg Cantor developed a method to prove a set is uncountable.
For example:
The set of all real numbers in the open interval
The intuition of the proof (by contradiction) is as follows:
Thus, the set of reals in
Suppose we denote a real number as
Index | Digits | |
---|---|---|
0 | ||
1 | ||
2 | ||
Next, we define a real number
We can see
Thus,
Here we list some facts about infinite sets:
There is no set of cardinality strictly between
So far, people only know:
CH is unprovable under ZFC.