高中数学 - 集合
2023-08-11 14:20:59 39 举报AI智能生成
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大纲/内容
6. 集合间的基本关系
包含关系”子集“
对于两个集合<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>，如果集合<span class="equation-text" data-index="2" data-equation="A" contenteditable="false"><span></span><span></span></span>中任意一个元素都是集合<span class="equation-text" data-index="3" data-equation="B" contenteditable="false"><span></span><span></span></span>的元素，那么集合<span class="equation-text" data-index="4" data-equation="A" contenteditable="false"><span></span><span></span></span>叫做集合<span class="equation-text" data-index="5" data-equation="B" contenteditable="false"><span></span><span></span></span>的子集，记作 <span class="equation-text" data-index="6" data-equation="A⊆B" contenteditable="false"><span></span><span></span></span>（或 <span class="equation-text" data-index="7" data-equation="B⊇A" contenteditable="false"><span></span><span></span></span>） ，<br>读作“<span class="equation-text" data-index="8" data-equation="A" contenteditable="false"><span></span><span></span></span>含于<span class="equation-text" data-index="9" data-equation="B" contenteditable="false"><span></span><span></span></span>”（“或<span class="equation-text" data-index="10" data-equation="B" contenteditable="false"><span></span><span></span></span>包含<span class="equation-text" contenteditable="false" data-index="11" data-equation="A"><span></span><span></span></span>”）
相等关系
集合<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>是集合<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>的子集，且<span class="equation-text" data-index="2" data-equation="B" contenteditable="false"><span></span><span></span></span>是<span class="equation-text" data-index="3" data-equation="A" contenteditable="false"><span></span><span></span></span>的子集<span class="equation-text" contenteditable="false" data-index="4" data-equation="（A\subseteq B且B\subseteq A）"><span></span><span></span></span>。此时<span class="equation-text" data-index="5" data-equation="A" contenteditable="false"><span></span><span></span></span>和<span class="equation-text" data-index="6" data-equation="B" contenteditable="false"><span></span><span></span></span>的元素是一样的，集合<span class="equation-text" data-index="7" data-equation="A" contenteditable="false"><span></span><span></span></span>等于集合<span class="equation-text" data-index="8" data-equation="B" contenteditable="false"><span></span><span></span></span>，记作<span class="equation-text" data-index="9" data-equation="A=B" contenteditable="false"><span></span><span></span></span>
真子集
对于两个集合<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>，如果<span class="equation-text" data-index="2" data-equation="A⊆B" contenteditable="false"><span></span><span></span></span>，并且<span class="equation-text" data-index="3" data-equation="A" contenteditable="false"><span></span><span></span></span>不等于<span class="equation-text" data-index="4" data-equation="B" contenteditable="false"><span></span><span></span></span>，我们就说集合<span class="equation-text" data-index="5" data-equation="A" contenteditable="false"><span></span><span></span></span>是集合<span class="equation-text" data-index="6" data-equation="B" contenteditable="false"><span></span><span></span></span>的<b>真子集</b>，记作<span class="equation-text" contenteditable="false" data-index="7" data-equation="A⫋B"><span></span><span></span></span>
空集
不含任何元素的集合称为空集，记作 <span class="equation-text" contenteditable="false" data-index="0" data-equation="∅"><span></span><span></span></span>
空集是任何集合的子集，记作<span class="equation-text" contenteditable="false" data-index="0" data-equation="∅\subseteq A"><span></span><span></span></span>
7. 集合的基本运算
子集
如果集合<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>的任意一个元素都是集合<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>的元素，那么集合<span class="equation-text" data-index="2" data-equation="A" contenteditable="false"><span></span><span></span></span>称为集合<span class="equation-text" data-index="3" data-equation="B" contenteditable="false"><span></span><span></span></span>的子集，若<span class="equation-text" data-index="4" data-equation="a∈A" contenteditable="false"><span></span><span></span></span>，均有<span class="equation-text" data-index="5" data-equation="a∈B" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" contenteditable="false" data-index="6" data-equation="A⊆B"><span></span><span></span></span>
交集
集合论中，设<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>是两个集合，由所有属于集合<span class="equation-text" data-index="2" data-equation="A" contenteditable="false"><span></span><span></span></span>且属于集合<span class="equation-text" data-index="3" data-equation="B" contenteditable="false"><span></span><span></span></span>的元素所组成的集合，叫做集合<span class="equation-text" data-index="4" data-equation="A" contenteditable="false"><span></span><span></span></span>与集合<span class="equation-text" data-index="5" data-equation="B" contenteditable="false"><span></span><span></span></span>的交集，记作<span class="equation-text" data-index="6" data-equation="A∩B" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="A∩B = B∩A" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(A∩B)∩C = A∩(B∩C)" contenteditable="false"><span></span><span></span></span>
并集
给定两个集合<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>，把他们所有的元素合并在一起组成的集合，叫做集合<span class="equation-text" data-index="2" data-equation="A" contenteditable="false"><span></span><span></span></span>与集合<span class="equation-text" data-index="3" data-equation="B" contenteditable="false"><span></span><span></span></span>的并集，记作<span class="equation-text" data-index="4" data-equation="A∪B" contenteditable="false"><span></span><span></span></span>，读作<span class="equation-text" data-index="5" data-equation="A" contenteditable="false"><span></span><span></span></span>并<span class="equation-text" contenteditable="false" data-index="6" data-equation="B"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A∪B = B∪A"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(A∪B)∪C = A∪(B∪C)" contenteditable="false"><span></span><span></span></span>
相对补集
若<span class="equation-text" data-index="0" data-equation="A" contenteditable="false"><span></span><span></span></span>和<span class="equation-text" data-index="1" data-equation="B" contenteditable="false"><span></span><span></span></span>是集合且<span class="equation-text" contenteditable="false" data-index="2" data-equation="A\subseteq B"><span></span><span></span></span>，则<span class="equation-text" data-index="3" data-equation="A" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="4" data-equation="B" contenteditable="false"><span></span><span></span></span>中的相对补集是这样一个集合：其元素属于<span class="equation-text" data-index="5" data-equation="B" contenteditable="false"><span></span><span></span></span>但不属于<span class="equation-text" data-index="6" data-equation="A" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="7" data-equation="B - A = \lbrace x| x∈B且x∉A\rbrace" contenteditable="false"><span></span><span></span></span>
绝对补集
若给定全集<span class="equation-text" data-index="0" data-equation="U" contenteditable="false"><span></span><span></span></span>，有<span class="equation-text" data-index="1" data-equation="A⊆U" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" data-index="2" data-equation="A" contenteditable="false"><span></span><span></span></span>在<span class="equation-text" data-index="3" data-equation="U" contenteditable="false"><span></span><span></span></span>中的相对补集称为<span class="equation-text" contenteditable="false" data-index="4" data-equation="A"><span></span><span></span></span>的绝对补集（或简称补集）
笛卡尔积（有序对）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A*B = \lbrace (a,b) \lvert a \in b, b \in a\rbrace"><span></span><span></span></span>
集合<span class="equation-text" data-index="0" data-equation="A = \lbrace 1,2,3 \rbrace" contenteditable="false"><span></span><span></span></span><br>集合<span class="equation-text" data-index="1" data-equation="B = \lbrace 6,66 \rbrace" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="2" data-equation="A*B = \lbrace (1,6),(2,6),(3,6),(1,66),(2,66),(3,66)\rbrace" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" contenteditable="false" data-index="3" data-equation="B*A = \lbrace (6,1),(6,2),(6,3),(66,1),(66,2),(66,3)\rbrace"><span></span><span></span></span><br>
其他运算
<span class="equation-text" data-index="0" data-equation="A∩(B∪C) = (A∩B)∪(A∩C)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="A∪(B∩C) = (A∪B)∩(A∪C)" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\overline{A∪B} = \overline{A} ∩ \overline{B}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\overline{A∩B} = \overline{A} ∪ \overline{B}" contenteditable="false"><span></span><span></span></span>
8. 区间
闭区间
闭区间表示一个范围内的所有元素，包括区间的两个端点。例如，闭区间 <span class="equation-text" data-index="0" data-equation="[a, b]" contenteditable="false"><span></span><span></span></span> 表示所有满足 <span class="equation-text" contenteditable="false" data-index="1" data-equation="a ≤ x ≤ b"><span></span><span></span></span> 的元素
开区间
开区间表示一个范围内的元素，但不包括区间的两个端点。例如，开区间<span class="equation-text" data-index="0" data-equation=" (a, b) " contenteditable="false"><span></span><span></span></span>表示所有满足<span class="equation-text" contenteditable="false" data-index="1" data-equation=" a < x < b "><span></span><span></span></span>的元素
半开半闭区间
这种区间一侧包含端点，另一侧不包含。例如，左闭右开区间<span class="equation-text" data-index="0" data-equation=" [a, b) " contenteditable="false"><span></span><span></span></span>表示所有满足 <span class="equation-text" contenteditable="false" data-index="1" data-equation="a ≤ x < b"><span></span><span></span></span> 的元素
无限区间
当区间延伸到无穷大时，可以使用 <span class="equation-text" data-index="0" data-equation="∞" contenteditable="false"><span></span><span></span></span> 表示。例如，开区间 <span class="equation-text" data-index="1" data-equation="(a, ∞)" contenteditable="false"><span></span><span></span></span> 表示所有大于 <span class="equation-text" contenteditable="false" data-index="2" data-equation="a"><span></span><span></span></span> 的实数
半无限区间
当区间在一侧延伸到无穷大时，可以使用 <span class="equation-text" contenteditable="false" data-index="0" data-equation="∞"><span></span><span></span></span> 表示。例如，左开右闭区间 <span class="equation-text" data-index="1" data-equation="(-∞, b]" contenteditable="false"><span></span><span></span></span> 表示所有小于等于<span class="equation-text" data-index="2" data-equation=" b" contenteditable="false"><span></span><span></span></span> 的实数
9. 无穷
在数学中，无穷是一个表示无限大的概念。例如，正整数的集合<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lbrace 1, 2, 3, ...\rbrace"><span></span><span></span></span>是一个无穷集合，因为它没有终止点。<br>然而，无穷并不是一个集合，而是一个描述性的概念，用来表示数量的无限性
无穷的运算
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(+∞)+(+∞)=+∞"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(+∞)-(+∞) 不确定"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(-∞)-(-∞)不确定"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(-∞)+(-∞)=-∞"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(+∞)*(+∞)=+∞" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(-∞)*(-∞)=+∞" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="(+∞)\div(+∞)不确定" contenteditable="false"><span></span><span></span></span>
10. 邻域
用来描述某个点周围的一组数。在实数集合中，给定一个实数 x，邻域可以表示为 <span class="equation-text" data-index="0" data-equation="(x - ε, x + ε)" contenteditable="false"><span></span><span></span></span>，其中 <span class="equation-text" data-index="1" data-equation="ε " contenteditable="false"><span></span><span></span></span>是一个正数，表示邻域的大小。<br>这个邻域包含了距离 <span class="equation-text" contenteditable="false" data-index="2" data-equation="x"><span></span><span></span></span> 不超过 <span class="equation-text" data-index="3" data-equation="ε " contenteditable="false"><span></span><span></span></span>的所有实数。
点<span class="equation-text" data-index="0" data-equation="a" contenteditable="false"><span></span><span></span></span>的<span class="equation-text" contenteditable="false" data-index="1" data-equation="δ"><span></span><span></span></span>邻域
设<span class="equation-text" data-index="0" data-equation="δ" contenteditable="false"><span></span><span></span></span>是一个正数，则开区间<span class="equation-text" data-index="1" data-equation="(a-δ，a+δ)" contenteditable="false"><span></span><span></span></span>称为点<span class="equation-text" data-index="2" data-equation="a" contenteditable="false"><span></span><span></span></span>的<span class="equation-text" contenteditable="false" data-index="3" data-equation="δ"><span></span><span></span></span>邻域，记作<br><span class="equation-text" data-index="4" data-equation="U(a, \delta) = \lbrace x | a - \delta < x < a + \delta \rbrace" contenteditable="false"><span></span><span></span></span><br>点<span class="equation-text" data-index="5" data-equation="a" contenteditable="false"><span></span><span></span></span>称为这个邻域的中心，<span class="equation-text" data-index="6" data-equation="δ" contenteditable="false"><span></span><span></span></span>称为这个邻域的半径<br>
去心邻域
以点 <span class="equation-text" data-index="0" data-equation="a" contenteditable="false"><span></span><span></span></span> 为中心，半径为 <span class="equation-text" data-index="1" data-equation="" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="2" data-equation="x" contenteditable="false"><span></span><span></span></span> 的一个开区间，排除了点 <span class="equation-text" data-index="3" data-equation="a" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="4" data-equation="" contenteditable="false"><span></span><span></span></span> 本身。<br>这个去心邻域包含了所有与点 <span class="equation-text" data-index="5" data-equation="a" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="6" data-equation="" contenteditable="false"><span></span><span></span></span> 的距离在 0 到 <span class="equation-text" data-index="7" data-equation="\delta" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="8" data-equation="" contenteditable="false"><span></span><span></span></span> 之间的点 <br><br><span class="equation-text" data-index="9" data-equation="" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="10" data-equation="\overset\wedge U (a,δ)=\lbrace x∣0<|x−a|<δ \rbrace" contenteditable="false"><span></span><span></span></span>
1. 集合的有关概念
一般地我们把研究对象统称为元素，把一些元素组成的总体叫集合(简称“集”)
构成两个集合的元素是一样的，则称这两个集合是相等的
2. 集合元素的特性
确定性：集合确定，则一个元素是否属于这个集合是确定的
互异性：一个给定集合中的元素是唯一的，不可重复
无序性：集合中元素的位置是可以改变的，改变位置不影响集合
3. 集合的表示方法
列举法 - 将集合中的元素一一列举出来
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lbrace a,b,c.....\rbrace"><span></span><span></span></span>
描述法 - 将集合中元素的公共属性描述出来，写在大括号内表示集合
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lbrace x \lvert x=5n+2,n∈Z\rbrace"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\lbrace a \lvert 0≤a≤10,a为奇数\rbrace" contenteditable="false"><span></span><span></span></span>
集合的表示通常用字母大写
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A,B,C"><span></span><span></span></span>
集合中的元素通常用字母小写表示
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a,b,c"><span></span><span></span></span>
某元素属于某集合
<span class="equation-text" contenteditable="false" data-index="0" data-equation="a∈A"><span></span><span></span></span>
某元素不属于某集合
<span class="equation-text" data-index="0" data-equation="a∉A" contenteditable="false"><span></span><span></span></span>
4. 集合的分类
有限集
有限个元素的集合
无限集
无限个元素的集合
空集
不含任何元素的集合
5. 集合中特殊数集的表示方法
实数集：<span class="equation-text" contenteditable="false" data-index="0" data-equation="R"><span></span><span></span></span>
正实数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="R^+"><span></span><span></span></span>
负实数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="R^-"><span></span><span></span></span>
零之外的：<span class="equation-text" contenteditable="false" data-index="0" data-equation="R^*"><span></span><span></span></span>
自然数集：<span class="equation-text" contenteditable="false" data-index="0" data-equation="N"><span></span><span></span></span>
正自然数集：<span class="equation-text" data-index="0" data-equation="N^+" contenteditable="false"><span></span><span></span></span>
零之外的：<span class="equation-text" data-index="0" data-equation="N^*"><span></span><span></span></span>
整数集：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Z"><span></span><span></span></span>
正整数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Z^+"><span></span><span></span></span>
负整数：<span class="equation-text" data-index="0" data-equation="Z^-" contenteditable="false"><span></span><span></span></span>
零之外的：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Z^*"><span></span><span></span></span>
正整数集：<span class="equation-text" data-index="0" data-equation="N^+或Z^+" contenteditable="false"><span></span><span></span></span>
有理数集：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q"><span></span><span></span></span>
正有理数：<span class="equation-text" data-index="0" data-equation="Q^+" contenteditable="false"><span></span><span></span></span>
负有理数：<span class="equation-text" data-index="0" data-equation="Q^-" contenteditable="false"><span></span><span></span></span>
零之外的：<span class="equation-text" contenteditable="false" data-index="0" data-equation="Q^*"><span></span><span></span></span>
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