---
title: 離散 part5
------
title: 離散 part5
---
{%hackmd @NTNUCSIE112/DM_card %}

# Set and functions
Ch.2 Basic Structure: Sets, Functions, Sequences, Sums, and Matrices

Reading assigned: 2.1, 2.2.1-2.2.3, 2.3.1-2.3.3, 2.5

## Cardinality (size) of a set (集合的元素個數)

The number of elements in a set S is called the size/cardinality of S,
denoted by |S|.

- e.g. S = {1, 2, 3, 4} --> |S| = 4.

Q: What is the cardinality of ? (Ans. ?)

Q: What is the cardinality of ? ? ? (Ans. ?)

Cantor (186x -- double check --)

- Consider and . -- , --> |N| < |Z|?

Q: why |N| = |Z|?

## Function

A founction: $f: S\to T$ is a "specific" mapping between element of $S$ and $T$ is such a way that "every element of $S$" corresponds to "an" element of $T$.
e.g.


## Set
### Definition (Countable set).
A set S is countable if S is finite or |S|=|N|.(If S is infinite, we say that S contains countably infinite elements)

### Theorem
Any subset of a countable set is countable
> 舉出所有有理數

<!-- 以下為課本(原文)章節 reading assignment 的 -->
## 2.1 Set
### 2.1.1 Introduction
### 2.1.2 Venn Diagrams
### 2.1.3 Subsets
### 2.1.4 The Size of a Set
### 2.1.5 Power Sets
### 2.1.6 Cartesian Products
### 2.1.7 Using Set Notation with Quantifiers
### 2.1.8 Truth Sets and Quantifiers
## 2.2

### 2.2.1
### 2.2.2
### 2.2.3
## 2.3
### 2.3.1
### 2.2.2
### 2.3.3
## 2.5
### 2.5.1
### 2.5.2 Countable Set
#### Hilbert's Grand Hotel

[TED - youtube](https://www.youtube.com/watch?v=Uj3_KqkI9Zo)
The Grand Hotel has infinite number of rooms. The rooms are numbered
from 1, 2, 3, ..., and each room is occupied by a guest.

-- A new guest arrives.

Q: Can you accommodate the new guest at the Grand Hotel?
### 2.5.3