---
# System prepended metadata

title: 離散數學

---

# 離散數學

## 集合
* Definition 1
    A set is an unordered collection of object

```python=
a = {1, 3, 5} # is a set
b = {4, 3, 5} # is a set
c = {1, 5, 3} # is a set

a = c # 無順序性
```

* Definition 2
    The object in a set are colled, the element, or the members, if the set.
    A set is said to contain ite element.
    If an element x is belong to a set A, then we denote it by x ∈ A
```python=
o = {1, 2, 3, 4, 5}
o = {x | .......} # => is range(1, 5) //


# another way to describe a set is to use set builder notation.
a = {1, 3, 5, 7, 9}
o = {x | is an odd positive intger less than 10}



o = {x ∈ z+ | x is odd and x < 10} 
```
* other example
```python=
# The set of integers:
z = {...-2, 0, ....}

# The set of positive integers:
z+ = {1, 2, 3...}

# The set of non-negative intergers:
z+(加號底下有0) = {0, 1, 2, 3}

# The set of rational numbers
Q = {p / q | p ∈ Z, q ∈ Z, and Q != 0}
```
![](https://i.imgur.com/eIYU8tg.png)


* Definition 3
    Two sets are equal
![](https://i.imgur.com/9GORYpU.png)
:::danger
if only if they have the same element. That is, if A and B are sets, than A and B are equal if and only if
:::


* 集合
![](https://i.imgur.com/DSiAVgw.png)
![](https://i.imgur.com/GFpXlGu.png)
![](https://i.imgur.com/QQQtntE.png)
![](https://i.imgur.com/Pxcwowr.png)
![](https://i.imgur.com/Yt2we3b.png)
* Definition 7
![](https://i.imgur.com/Dkj3KUF.png)
![](https://i.imgur.com/fHfapKA.png)
![](https://i.imgur.com/eUKDOFQ.png)
![](https://i.imgur.com/YuZBTfM.png)
![](https://i.imgur.com/9GOUJx4.png)
![](https://i.imgur.com/1mLW0hX.png)
* 0![](https://i.imgur.com/gIiT0WA.png)
![](https://i.imgur.com/0CtlbJu.png)
![](https://i.imgur.com/TR4pvVS.png)

* example 7
![](https://i.imgur.com/J3Ef2iJ.png)
* one - to - one (到這)
![](https://i.imgur.com/aqXDloN.png)
![](https://i.imgur.com/pKorbbM.png)

* example 8
![](https://i.imgur.com/45O0e75.png)

* example 9
![](https://i.imgur.com/VqxTDrs.png)

* note
![](https://i.imgur.com/FucwQo4.png)
* &&
![](https://i.imgur.com/n95IxRz.png)

* one - to - one
![](https://i.imgur.com/2pSEkYR.png)
![](https://i.imgur.com/L65I1q1.png)

* defination 6
![](https://i.imgur.com/MmfBSfz.png)
* 廣義遞增
![](https://i.imgur.com/7mDtq3y.png)
* 遞增遞減
![](https://i.imgur.com/I1lstDm.png)

* defination 7
![](https://i.imgur.com/QVwuV39.png)

* example 12
![](https://i.imgur.com/FYbYVQb.png)

* example 13(使用定義  by defination onto function)
![](https://i.imgur.com/HgwSjP9.png)

* example 14(not onto)
![](https://i.imgur.com/zkdFQwI.png)


* defination  8
![](https://i.imgur.com/syeuDsO.png)


* example 16 
![](https://i.imgur.com/jAx7gHm.png)


* one - hence
![](https://i.imgur.com/khuepBn.png)

* note
![](https://i.imgur.com/7CkSY14.png)

* defination 9
![](https://i.imgur.com/lWKboOB.png)

* 嚴謹寬鬆
![](https://i.imgur.com/eLUnGWX.png)

* 合成函數與連鎖率
![](https://i.imgur.com/vfSj8wN.png)

* 2.5
![](https://i.imgur.com/FreD3lB.png)
* defination 2
![](https://i.imgur.com/Ij08wo0.png)
* 證明無窮可數
![](https://i.imgur.com/77EtkcU.png)
* example 3
![](https://i.imgur.com/2BnPp84.png)
* 可數
![](https://i.imgur.com/VnKyQpm.png)
* example 4
![](https://i.imgur.com/d6sSZC8.png)
* 數羊
![](https://i.imgur.com/xek4giW.png)
![](https://i.imgur.com/95FWWwd.png)
![](https://i.imgur.com/SwyVGZT.png)
![](https://i.imgur.com/sBcxds3.png)
## chapter 3
![](https://i.imgur.com/lCs6aWV.png)
* example 1
![](https://i.imgur.com/bZuCs9N.png)
* 0.0
![](https://i.imgur.com/WKIvxQB.png)
* example 3
![](https://i.imgur.com/PpJLSFk.png)
* example 5
![](https://i.imgur.com/kBfqdxn.png)
![](https://i.imgur.com/XjPPnU3.png)
![](https://i.imgur.com/go53UcG.png)
![](https://i.imgur.com/5ZBHXvu.png)
![](https://i.imgur.com/Qut0ANE.png)
![](https://i.imgur.com/AIHGDVq.png)
![](https://i.imgur.com/f0DJscF.png)
![](https://i.imgur.com/y7MVibp.png)


