2/23/2004
Discrete Mathematics for Teachers, UT Math 504, Lecture 07
Cardinality, Recursion, and Matrices
Sections 4.3-4.4, 5.2, 5.8
Cardinality
It was the work of Georg Cantor (1845–1918) to establish the field of set theory and to discover that infinite sets can have different sizes. His work was controversial in the beginning but quickly became a foundation of modern mathematics. Cardinality is the general term for the size of a set, whether finite or infinite.
Definition for finite sets
The cardinality of a finite set is simply the number of elements in the set. For instance the cardinality of {a,b,c} is 3 and the cardinality of the empty set is 0. More formally the set A has cardinality n (for nonnegative integer n) if there is a bijection (one-to-one correspondence) between the set {1,2,3,…,n} and the set A. Formally then we define the bijection f:{1,2,3}→{a,b,c} by f(1)=a, f(2)=b, and f(3)=c, thereby proving {a,b,c} has cardinality 3.
Definition for finite sets
Although our book saves the notation for later, we denote the cardinality of the set A by |A|. This looks like "absolute value" and it measures the size of a set, just as absolute value measures the magnitude of a real number. Another useful notation that does not appear in our book is to let [n] be the set {1,2,3,…n} with [0]= . Then we can state that |A|=n if there is a bijection f:[n]→A.
Definition of countable
An infinite set A is countably infinite if there is a bijection f: →A, where is the set of positive integers. That is ={1,2,3,…}.
A set is countable if it finite or countably infinite. A synonym for countable is denumerable. Infinite sets that are not countable are uncountable or, less frequently, nondenumerable.
Elementary Theorems
The cardinality of the disjoint union of finite sets is the sum of the cardinalities (4.45a). That is, suppose |A|=m and |B|=n for nonnegative integers m and n and disjoint sets A and B. Then |A∪B|=m+n. Proof. There exist bijections f:[m]→A and g:[n]→B. Define a function h:[m+n]→(A∪B) by h(x)=f(x) if 1≤x≤m and h(x)=g(x–m) if m+1≤x≤m+n. It is tedious but not hard to show that f is a bijection. The following picture makes the situation clear.
- 古加尔25人 > cardinalityrecursionand matrices sections 4.3-4.45.25.8
-
cardinalityrecursionand matrices sections 4.3-4.45.25.8
下载该文档 文档格式:PPT 更新时间:2004-06-01 下载次数:0 点击次数:1文档基本属性 文档语言: 文档格式: ppt 文档作者: Reid Davis 关键词: 主题: 备注: 点击这里显示更多文档属性 经理: 单位: The University of Tennessee 分类: 创建时间: 上次保存者: Reid Davis 修订次数: 105 编辑时间: 文档创建者: 修订: 加密标识: 幻灯片: 53 段落数: 169 字节数: 185009 备注: 0 演示格式: On-screen Show 上次保存时间:
- 下载地址 (推荐使用迅雷下载地址,速度快,支持断点续传)
- PPT格式下载
- 更多文档...
-
上一篇:年10月25日
下一篇:广东省2010高中青年教师命题大赛(25)
点击查看更多关于古加尔25人的相关文档
- 您可能感兴趣的
- 古加尔 大脚论坛古加尔 h古加尔攻略 特古西加尔巴 10h古加尔攻略 迅捷微风古加尔首杀 古加尔身上的眼睛 10h古加尔 特古西加尔巴机场
- 大家在找
-
- · yy小说下载txt电子书
- · ssd安装xp
- · 川教版初一语文
- · 望庐山瀑布古诗配画
- · 钳工技术比武考试题
- · 电工实验教案
- · 海聊视频聊天
- · 组织代码上海市建工设计研究院有限公司党委
- · 管壳式换热器结构图
- · qq餐厅仓库位置
- · 塔吊电路图下载
- · 无锡加工中心操作工
- · 江苏新众亚货架制造有限公司
- · 连发程序威盘
- · 盘锦高铁建设
- · 西游记张卫健版
- · 高二政治哲学试题
- · 司机招聘网
- · 如何做一名优秀的教师
- · 广西大学函授本科报名
- · www.ems.com.con
- · dnf蜗牛1.06下载
- · 求购pcb线路板
- · 长江七号爱地球全集
- · mx有没有前置摄像头
- · 湖南邵阳武冈公安局
- · 外科护理学试题答案
- · 国际贸易理论与实务论文
- · plc编程实例108
- · 2012年贷款卡年审表格
- 赞助商链接