课程： 1.2 一套 | The Way To Learn

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In the first half of 2014, the Centers for Disease Control and Prevention (CDC) collected data on whether households had landline phones or used cell phones for their phone service. They found that 44% of households only used cell phones, 8.5% of households only used a landline, 2.6% of households had no phone service, and they were unable to obtain information from 0.2% of households. ¹
::在2014年上半年,疾病控制和预防中心收集了家庭是否拥有固定电话或使用移动电话提供电话服务的数据,发现44%的家庭只使用手机,8.5%的家庭只使用固定电话,2.6%的家庭没有电话服务,无法从0.2%的家庭获得信息。

By treating each of these categories as what is called a set, we also can determine how many households used landlines and cell phones. We will discuss sets in this section.
::通过将这些类别中的每一个类别视为所谓的一套,我们也可以确定有多少家庭使用陆上线路和手机,我们将在本节讨论各组。

Sets
::套套套套套套

A set is a collection of objects called elements . We use curly brackets, { and }, to enclose the elements in a set or the description of the set. We label sets with capital letters. For example, the set A={red, orange, yellow, green, blue, indigo, violet} is the set of colors in a rainbow or the set B={the states in the U.S.} is the set of all fifty states. When we describe the elements in a set by listing them, as in set A, it is referred to as the roster method .
::一组是称为元素的物体集合。我们使用卷括号 { 和} 来将元素附加在一组或一组描述中。我们用大写字母来标注组。例如, 一组 A red、橙、黄色、绿色、绿色、蓝色、 indigo、紫罗兰} 是彩虹中的颜色组, 或一组 B 美国州的颜色组 , 是全部五十个州的集合。当我们在一组中描述元素时, 如一组 A 中列出它们时, 它被称为名册方法。

The number of elements in a set is called the cardinality of the set. The cardinality of set A is denoted $\begin{aligned} | A | \end{aligned}$ . A set that has a limited number of elements is called a finite set. Set A above has 7 elements. It is finite. If the set has an unlimited number of elements, it is called an infinite set. For example, the set of points in space or the set of all multiples of three are infinite sets.
::一组中元素的数被称为集的基数。集 A 的基数表示 A 。数量有限的元素的组数称为限数的组数。上面的 A 有 7 个元素。如果集有无限数量的元素, 它被称为无限的组数。例如, 空间中的一组点数或所有三个数的组数都是无限的组数。

If the set has no elements, it is called the empty set or the null set . We denote the empty set either with empty brackets, { }, or ∅.
::如果集没有元素, 则称为空套件或空套件。我们用空括号 {} 或\\\ 来表示空套件。

Example 1
::例1

Determine the cardinality of the following sets and state whether they are infinite or finite.
::确定以下各组的基点,并说明它们是无限的还是有限的。

a. A={1,2,3,4,5,6,7,8,9,10}
::a. A1,2,3,4,5,6,7,7,8,9,9,10}

b. B={2,4,6,8,10,...} (... means the pattern continues forever)
::b. B2,4,6,8,10,......}(.意指这种模式将永远持续下去)

c. C={colors that are numbers}
::c. 数字的Ccolor}

Solution:
::解决方案 :

a. $\begin{aligned} | A | \end{aligned}$ = 10. It is a finite set.
::a. A=10。

b. The set B of even numbers is infinite.
::b. 偶数的B组数是无限的。

c. Since there are no colors that are also numbers, the set C has no elements. It is an empty or null set, that is, $\begin{aligned} C = \emptyset \end{aligned}$ or $\begin{aligned} C = {} \end{aligned}$ . Thus, $\begin{aligned} | C | = 0 \end{aligned}$ . It is a finite set.
::c. 由于没有也为数字的颜色,设置的C没有元素。它是一个空或空的集合,即C或C。因此,这是个限定的集合。

by Mathispower4u introduces some vocabulary used for sets.
::由 Mathispower4u 介绍一些用于集的词汇。

Subsets
::子设置

A subset is a set that has all of its elements also contained in another set. The notation to indicate that one set is a subset of another is $\begin{aligned} \subseteq \end{aligned}$ . For example, $\begin{aligned} C = {1, 2, 3} \end{aligned}$ and $\begin{aligned} D = {1, 2, 3, 4, 5} \end{aligned}$ , so $\begin{aligned} C \subseteq D \end{aligned}$ . However, C is also an example of a proper subset of D since it does not contain all of the elements in D, just some of them. To indicate that C is a proper subset of D, w e write $\begin{aligned} C \subset D \end{aligned}$ . Not all subsets are proper subsets however. If $\begin{aligned} A = {1, 2, 3, 4, 5} \end{aligned}$ , we could say that $\begin{aligned} A \subseteq D \end{aligned}$ .
::子集是一个包含全部元素的集。表示一组元素是另一组元素的一个子集。表示一组元素是另一组元素的一个子集的注解是。例如, C1, 2, 3} 和 D1, 2, 3,4,5}, 所以是 CD。然而, C 也是D 的适当子集的一个例子, 因为它并不包含 D 中的所有元素, 只是其中的一些元素。为了表明 C 是 D 中的一个适当的子集, 我们写了 CD 。但是, 不是所有子集都是适当的子集。如果 A1, 2, 3,4,5}, 我们可以说 AD 。

If $\begin{aligned} F = {7, 8, 9} \end{aligned}$ , F is not a subset of D because none of the elements in F are contained in D. S ymbolically, we would indicate this as $\begin{aligned} F ⊈ D \end{aligned}$ ,
::如果F+7,8,9},F不是D的子集,因为F中的任何元素都没有包含在D。

The empty set is considered a subset of all sets.
::空套件被视为所有套件的子集。

If two sets contain the same elements, like A and D above, they are considered equal . We would say $\begin{aligned} D = A \end{aligned}$ . If two sets have the same cardinality, they are considered equivalent . The symbol that represents equivalent is $\begin{aligned} ≅ \end{aligned}$ . Sets that are equal are also equivalent. However, sets may be equivalent, but not equal. For example, if $\begin{aligned} G = {a, b, c, d, e} \end{aligned}$ , then $\begin{aligned} D ≅ G \end{aligned}$ , but $\begin{aligned} D \neq G \end{aligned}$ .
::如果两组包含相同的元素, 如上面的 A 和 D 一样, 它们被认为是相等的。我们会说 D = A 。如果两组具有相同的基本特征, 它们就被认为是等效的。表示等值的符号是 \\ 。等值的符号是 \ 。等值的字符也相等。但是, 组可以是等值的, 但不是等值的。例如, 如果 G , b, c, d, e}, 那么 D = G , 但 D + G 。

We can also determine the number of subsets of a set. If a set has n elements , it will have $\begin{aligned} 2^{n} \end{aligned}$ subsets and $\begin{aligned} 2^{n} - 1 \end{aligned}$ proper subsets. For example, for M={1,5,10}, n = 3, because $\begin{aligned} | M | = 3 \end{aligned}$ . M would have 8 subsets ( $\begin{aligned} 2^{3} = 8 \end{aligned}$ ) and 7 proper subsets (8 - 1 = 7).
::我们还可以确定一组子集的数量。如果一组具有 n 元素, 它将拥有 2n 子集和 2n-1 适当的子集。例如, 对于 M1, 5, 10}, n = 3, 因为 M 3 . M 将拥有 8 子集 (23=8) 和 7 个适当子集 (8-1 = 7) 。

by Mathispower4u defines and provides the notation used for subsets and proper subsets.
::由 Mathispower4u 定义并提供子集和适当子集使用的标记。

Complements
::补充

A universal set is the set of all of the elements in a given situation. Suppose some asks you to "pick a number between 1 and 10" (assuming the intention is to get a whole number as a response). The universal set is then, $\begin{aligned} U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} \end{aligned}$ .
::通用套件是特定情况下所有元素的集合。如果有人要求您“选择一个在1到10之间的数字 ” ( 假设意图是获得一个完整的数字作为回应 ) 。那么通用套件就是 U1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10} 。

Let's then suppose that you do not want to pick a number less than 5, or you want to pick numbers in the set $\begin{aligned} C = {5, 6, 7, 8, 9, 10} \end{aligned}$ . The complement of a set is all of the elements in the universal set not in the set itself. The complement of a set C can either be denoted as $\begin{aligned} \bar{C} \end{aligned}$ or as C'. In this example, $\begin{aligned} \bar{C} = {1, 2, 3, 4} \end{aligned}$ or all of the numbers you did not want to pick.
::假设您不想选择一个小于 5 的数, 或者您不想在设定的 C 5, 7, 8, 8, 9, 10} 中选择数字。一组的补全是全域集中的所有元素, 而不是集本身。一组 C 的补全可以被称为 C 或 C 。在此示例中, C 1, 2, 3,4} 或者您不想选择的所有数字。

Union and Intersection
::工会和跨部门组织

There are two operations between sets— union and intersection .
::结合和十字路口之间有两种操作。

Union and Intersection
::工会和跨部门组织

The union of two sets is the set that contains all of the elements in both sets. We use $\begin{aligned} \cup \end{aligned}$ to indicate the union of two sets.
The intersection of two sets is the set that contains elements that are in both sets, and we indicate it with $\begin{aligned} \cap \end{aligned}$ .
::两组的交叉点是包含两组中元素的集,我们用表示。

If the intersection of two sets is empty, then the sets are said to be disjoint .
::如果两组的交叉点是空的,则两组的交叉点据说是脱节的。

Example 2
::例2

Consider two sets A={a,l,g,e,b,r,a} and B={n,u,m,b,e,r}. Find the union and intersection of A and B.
::考虑两组Aa,l,g,e,b,r,a}和Bn,u,m,b,e,r}。找出A和B的结合和交叉点。

Solution:
The union is all of the letters in both sets. We do not have to list the repeated elements twice, like for b or e.
::解决方案: 联盟是两组中的所有字母。我们不必两次列出重复元素, 比如 b 或 e 。

$\begin{aligned} A \cup B \end{aligned}$ = {a,l,g,e,b,r,n,u,m}
::B = {a,l,g,e,b,r,r,r,n,u,m}

The intersection of the two sets is the elements or letters in both sets. e, b, and r are in both sets, so those elements are in the intersection of A and B.
::两组的交叉点是两组中的元素或字母。 e、b和r是两组中的元素或字母,因此这些元素位于A和B的交叉点。

$\begin{aligned} A \cap B \end{aligned}$ = {e,b,r}
::AB = {e,b,r} = {e,b,r} = {e,b,r} = {e,b,r} = {e,b,r}

by the Learning Assistance Center of Howard Community College explains intersection, union, and complements. It also shows a visual representation of each using .
::霍华德社区学院学习援助中心解释交叉点、联合点和补充点。

Venn Diagrams
::文文图

A is a pictorial representation of the relationships between sets. A general example is shown below.
::A是各组之间关系的图示,下面是一般例子。

The diagram illustrates that within a universal set of data (represented by the rectangle) , there are two sets, A and B (represented by the circles). These sets are not disjoint, because their intersection is not empty. They have some elements in common. Now let's consider an example with data.
::该图表显示,在一套通用数据(以矩形表示)中,有两组数据,A组和B组(以圆为代表),这些组并非脱节,因为它们的交叉点不是空的。它们有一些共同元素。现在让我们用数据来举一个例子。

Example 3
::例3

In the first half of 2014, the CDC found that 44% of households only used cell phones, 8.5% of households only used a landline, 2.6% of households had no phone service, and they were unable to obtain information from 0.2% of households. ¹ Create a Venn diagram to illustrate this information.
::2014年上半年,疾病防治中心发现,44%的家庭只使用手机,8.5%的家庭只使用固定电话,2.6%的家庭没有电话服务,他们无法从0.2%的家庭获得信息。 1 制作一个文恩图来说明这一信息。

Solution: First, let's label the parts that we know. In the part of the " L " circle that does not overlap with the " C " circle we put the landline only households or 8.5%. Similarly, we put 44% in the " C " circle where the two circles do not overlap. The households that had no phone service are not in either set L or C , but they are households, so are part of the universal set inside the box. We leave a note at the bottom of the diagram indicating the households for which the CDC was unable to obtain information.
::解决方案 : 首先, 让我们标出已知的部分。在“ L” 圆圈中与“ C” 圆圈不重叠的部分, 我们将“ L” 线只标为家庭或8.5% 。同样, 我们将44% 放在“ C” 圆圈中, 其中两个圆圈没有重叠。没有电话服务的家庭没有在 L 或 C 里设置, 但是它们是家庭, 也属于盒子里通用的组合的一部分。我们在图表的底部留下一张纸条, 说明CDC 无法获取信息的家庭。

The only part missing is the intersection. If we consider the universe 100% of the data, the intersection, that is households that use both landlines and cell phones, make up the missing part.
::唯一缺少的部分是交叉路口。如果我们把宇宙 100%的数据, 交叉路口,即使用地线和手机的家庭, 组成了缺失的部分。

$\begin{aligned} 100 - (44 + 8.5 + 2.6 + 0.2) = 100 - 55.3 = 44.7 \end{aligned}$

We can now include 44.7% where the two circles overlap. The completed Venn diagram is below.
::我们现在可以包括44.7%的两圈重叠的地方。完成的文恩图如下。

Example 4
::例4

Create a Venn diagram to represent the following information and answer the questions that follow.
::创建 Venn 图表以显示以下信息并回答以下问题。

In a survey of 150 students it was found that:
::在对150名学生的调查中发现:

80 student commute by bus
::80名学生乘坐公共汽车

110 students commute by car
::110名学生乘车往返

125 students commute by walking
::125名学生步行往返

62 students commute by both bus and car
::62名学生乘坐公共汽车和汽车

58 students commute by both bus and walking
::58名学生乘坐公共汽车和步行,

98 students commute by car and walking
::98名学生乘汽车上下班和步行

50 students commute in all three ways
::50名学生以所有三种方式通勤

a. How many students only commute by car?
::a. 有多少学生只乘车上学?

b. How many students commute by a different method?
::b. 有多少学生以不同方法通勤?

c. How many students commute by walking and taking the bus but not by a car?
::c. 有多少学生步行乘公交车而不是乘车往返?

Solution: First, we will use the given information to construct the Venn diagram. We can start by putting $\begin{aligned} 50 \end{aligned}$ in the center where students by all three methods . Next we can find the values in blue by subtracting 50 from the number of students that commute by two methods .
::解决方案 : 首先, 我们将使用给定的信息构建 Venn 图表。我们可以首先将50 个学生以所有三种方法放入中心。下一步, 我们可以从以两种方法通勤的学生人数中减去50个, 找到蓝色的值。

$\begin{aligned} bus and car & : 62 - 50 = 12 \\ bus and walking & : 58 - 50 = 8 \\ car and walking & : 98 - 50 = 48 \end{aligned}$

::公共汽车和汽车:62-50=12bus和行走:58-50=8car和行走:98-50=48

Once the blue values are found, we can find the green values by subtracting the blue and red values in each subset from the total in the subset.
$\begin{aligned} bus only & : 80 - (50 + 12 + 8) = 10 \\ car only & : 110 - (50 + 12 + 48) = 0 \\ walking only & : 125 - (50 + 8 + 48) = 19 \end{aligned}$

::一旦找到蓝色值,我们就可以从子集的总和中减去每个子集中的蓝色和红色值,找到绿色值。公共汽车: 80- (50+12+8) = 10car: 110- (50+12+48) = 0行走仅: 125- (50+8+48) = 19

Finally we can add up all the values in the circles and subtract this from 150, the total number of students surveyed to determine that 3 students do not commute any of these three ways.
::最后,我们可以将圈子中的所有数值加在一起,从150个学生中减去,即被调查的学生总数,以确定有3名学生不以上述三种方式通勤。

Now that the Venn diagram is complete, we can use it to answer the questions.
::现在文恩图已经完成了我们可以用它回答问题

a. There are 0 students that only commute by car.
::a. 有0名学生仅乘车往返。

b. There are 3 students that commute by another method .
::b. 有3名学生以另一种方式通勤。

c. There are 8 students that commute by the bus and by walking .
::c. 有8名学生乘坐公交车和步行。

Feature: European Union
::特点:欧洲联盟

by Deirdre Mundy
::由Deirdre Mundy 编辑

Why is the European Union (E.U.) made up of nations, instead of states like the United States? Why have some European countries not joined the E.U.? Why are there so many treaties that include some parts of the E.U. but not others? For an outside observer, the E.U. can seem confusing. It helps to understand how it grew, what it is today, and why some countries choose to remain outside of it.
::为什么欧洲联盟(E.U.)是由国家而不是美国这样的国家组成?为什么有些欧洲国家没有加入欧盟?为什么有这么多条约包括欧盟的某些部分,而没有包括其它部分?对外部观察者来说,欧盟似乎会混淆不清。它有助于理解欧盟是如何成长的,今天是什么,为什么某些国家选择不加入欧盟。

Coal, Steel, Trade and Peace
::煤炭、钢铁、贸易与和平

The E.U. has its roots in the years following World War II. In less than 40 years, the European nations endured two terrible, destructive wars that crippled their economies, decimated their populations, and scarred their landscapes. Leading nations saw trade unions and shared defenses as a way to prevent future wars, and to protect Europe from invasion by larger nations.
::在二战后的几年里,欧盟的根基在于二战。在不到40年的时间里,欧洲各国经历了两次可怕的毁灭性战争,这些战争使欧洲经济瘫痪,人口被毁灭,地貌被打成碎片。领导国家将工会和共同防御视为防止未来战争、保护欧洲免受大国入侵的一种方法。

The beginnings of the E.U. are in the European Coal and Steel Community, an agreement between Belgium, France, Germany, Italy, Luxemburg, and the Netherlands, was established in 1951. As time went on, membership requirements changed and more countries joined what was then called the European Common Market or the European Economic Community. In 1993, this group was incorporated into the newly formed European Union. In 2004, the E.U. responded to the fall of the Soviet Union and the economic rise of Eastern Europe. It allowed the former communist countries of the Czech Republic, Estonia, Latvia, Lithuania, Hungary, Poland, Slovakia and Slovenia to join.
::欧洲煤钢共同体是1951年比利时、法国、德国、意大利、卢森堡和荷兰签署的协定,随着时间的流逝,加入欧盟的要求发生了变化,更多的国家加入了当时称为欧洲共同市场或欧洲经济共同体的国家。1993年,该集团并入了新成立的欧洲联盟。2004年,欧盟对苏联的垮台和东欧的经济崛起作出反应,允许捷克共和国、爱沙尼亚、拉脱维亚、立陶宛、匈牙利、波兰、斯洛伐克和斯洛文尼亚的前共产主义国家加入。

Most of the countries of the E.U. share a single currency called the Euro. They also agree to limits on their economic freedom and ability to make independent treaties. The E.U. is also known for carefully regulating the products traded within its boundaries. An Euler diagram , similar to a Venn diagram, can help you understand which countries are part of which agreements, and how different trade agreements affect E.U. countries.
::大部分欧盟国家都拥有一种单一货币,称为欧元,它们也同意限制其经济自由和缔结独立条约的能力。欧共体也以谨慎监管其边界内交易的产品而著称。与文恩图类似的欧勒图可以帮助你理解哪些国家是其中一部分,以及不同的贸易协定如何影响欧盟国家。

Note: This image is from before the United Kingdom voted to leave the E.U.
::注:此图来自英国投票退出欧盟之前的英国。

Some European countries choose to remain outside of the E.U. Switzerland is not a member because its citizens value neutrality and independence. Norway has not joined the E.U. because its citizens feel that E.U. regulations would destroy both its fishing industry and its social programs. It has held two referendums on E.U. membership, but both times the people of Norway voted to remain separate from the E.U.
::一些欧洲国家选择不加入欧盟,因为其公民重视中立和独立。挪威没有加入欧盟,因为其公民认为欧盟的规章会破坏其渔业和社会方案。挪威举行了两次关于欧盟成员资格的全民投票,但挪威人民两次投票决定与欧盟分离。

by European Commission describes how E.U. trade works.
::欧盟委员会介绍了欧盟贸易如何运作。



Summary
::摘要

A set is a group of objects called elements.
::一组是一组名称为元素的物体。

::一组是一组名称为元素的物体。

The number of elements in a set is called the cardinality of the set and is denoted $\begin{aligned} | A | \end{aligned}$ .
::一组元素的编号称为集的基点,并标注 A。

::一组元素的编号称为集的基点,并标注 A。

The complement of a set is all of the elements that are not in the set, but are in the universal set, and is denoted A '.
::一组中的补充是非集中的所有要素,而是通用集中的所有要素,并指A'。

::一组中的补充是非集中的所有要素,而是通用集中的所有要素,并指A'。

The union of two sets is all of the elements that are in either set. The symbol for union is ∪.
::两组组合的结合是两组组合中的所有元素。工会的符号是。

::两组组合的结合是两组组合中的所有元素。工会的符号是。

The intersection of two sets is all of the elements in both sets. The symbol for intersection is ∩.
::两组的交叉点是两组中的所有元素。交叉点的符号是。

::两组的交叉点是两组中的所有元素。交叉点的符号是。

We can use Venn diagrams to represent the relationships between sets.
::我们可以使用文恩图表来代表各组之间的关系。

Review
::回顾

1. Use the roster method to represent M, the set of months of the year that begin with the letter ‘J.’ Then find |M|.
::1. 使用名册方法代表M,即从字母 " J. " 开始的一年中数月系列的M,然后找到 ' M. ' 。

2. Determine if the given pairs of sets are equal, equivalent, or neither. Find the cardinality of each of the sets.

::2. 确定给定的两组是否相等、等同或两者兼而有之。

a. F = {Jane, Jada, Janelle, Julie, Jessica, Jacinta} and M = {Joe, Jeff, John, Jamal, Justin, Jason}
::a. F={Jane、Jada、Janelle、Julie、Jessica、Jacinta}和M={Joe、Jeff、John、Jamal、Justin、Jason};和M={Joe、Jeff、John、Jamal、Justin、Jason};

b. X = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} and Y = {18, 14, 12, 6, 8, 4, 2, 10, 16, 20}
::b. X = {2,4,6,8,10,12,14,16,18,20}和Y = {18,14,12,6,8,8,4,2,10,16,20}

Find a subset and its complement for the following sets.
::为以下各组寻找子集及其补充。

3. {a, b, c, d, e}
::3. {a, b, c, d, d, e}

4. {physics, chemistry, biology, engineering, computer science, accounting, business, liberal arts}
::4. {物理、化学、生物、工程、计算机科学、会计、商业、自由艺术}

5. {Paul, Ringo, John, George}
::5.{保罗,林戈,约翰,乔治}

6. Let A= {m, a, t, h, e, m, a, t, i, c, s} and B = {c, o, l, l, e, g, e} and C = {a, l, g, e, b, r, a}.
::6. 让A={m, a, t, h, e, m, a, t, i, c, s}和B={c, o, l, l, e, g, e}和C={a, l, g, e, b, r, a}。

a. Find A ∪ B.
::a. 找到A _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

b. Find A ∩ B.
::b. 查找A _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

c. Find B ∩ C.
::c. 查找B _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

d. Find (A ∩ B) ∩ C.

::d. 查找(A {___________________________________________________________________________________________________________________

Use the letters, w, x, y, and z, in the Venn diagram below to describe the region(s) that represent each of the sets.
::使用下文文恩图中的字母w、x、y和z来描述代表每一组的区域。

7. A ∩ B
::7. A __________________________________________________________________________________________________________________________________

8. A
::8. A 级

9. A ∪ B
::9. A ____________________________________________________________B

10. A ∩ B'
::10. A B'

11. (A ∩ B)'
::11. (A B)'

12. (A ∪ B)'
::12. (A B)'

13. A'
::13. A'

14. B' ∪ A
::14. `````A'

Explore More
::探索更多

1. In a survey of 80 households asking whether the households had a dog, cat, or another pet, like a fish, turtle, reptile, hamster, etc. , it was found that:
::1. 对80户家庭进行调查,询问这些家庭是否有一只狗、一只猫或另一只宠物,如鱼、海龟、爬行动物、仓鼠等,结果发现:

30 had at least one dog
::30个有至少一只狗

42 had at least one cat
::42个有至少一只猫

21 had at least one other pet
::21只有至少另一只宠物

20 had dog(s) and cat(s)
::20只有狗和猫

10 had cat(s) and other pet(s)
::10只猫和其他宠物

8 had dog(s) and other pet(s)
::8只有狗和其他宠物

5 had all three types of pets
::5个有所有三种宠物的宠物

Make a Venn diagram to illustrate the results of the survey. How many have at least one dog and at least one cat but no other pets? How many have only a dog or dogs? How many have no pets at all? How many other pet owners also have at least one dog or cat but not both?
::用文恩图来说明调查结果。有多少人至少有一只狗和至少一只猫,但没有其他宠物?有多少人只有一只狗或狗?有多少人根本没有宠物?有多少其他宠物拥有至少一只狗或猫,但没有两者?

2. Freezy's Ice Cream Stand polls its customers on their favorite flavor: chocolate or vanilla? 103 customers said they liked chocolate, 98 customer said they like vanilla, while 27 customers said they liked both chocolate and vanilla. How many customers said they like only chocolate?
::2. Freezy的《冰淇淋》调查其客户的口味:巧克力或香草?103个客户说他们喜欢巧克力,98个客户说他们喜欢香草,27个客户说他们既喜欢巧克力,也喜欢香草。有多少客户说他们只喜欢巧克力?

3. The power set of a set is the set of all possible subsets of the set. For example, the power set of $\begin{aligned} {a, b, c} \end{aligned}$ is $\begin{aligned} {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} \end{aligned}$ .
::3. 集的功率组是集的所有可能的子集的集。例如, {a, b, c} 的功率组是{a} {b} {c} {c} {a, b} {a, c} {b, c} {a, b, c} {a, b, c} 。

Let X = {3, 4, 5, 6}. Find the power set of X.
::让 X = {3、4、 5、 6} 找到 X 的电源组。

4. Determine if the following sets are empty sets:
::4. 确定以下各组是否为空组:

a. The set N of negative numbers less than 20.
::a. 负数小于20的一组负数N。

b. The set M of months of the year that have 27 days
::b. 每年27天的月数M数的设定值

5. Below are DeMorgan's Laws, which are important in logic and algebra. Show that DeMorgan's Laws are true using Venn diagrams.

::5. 以下是DeMorgan的法律,这些法律在逻辑和代数方面很重要,用Venn图表显示DeMorgan的法律是真实的。

$\begin{aligned} (A \cup B)^{'} = A^{'} \cap B^{'} \\ (A \cap B)^{'} = A^{'} \cup B^{'} \end{aligned}$

:AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB) (AB)

6. Use a Venn diagram to describe a real-world situation as in the Feature article in this section.

::6. 使用文恩图来描述本节特写文章中的实际情况。

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

1. “Wireless Substitution: Early Release of Estimates From the National Health Interview Survey, January–June 2015,” by Stephen J. Blumberg, Ph.D., and Julian V. Luke, Division of Health Interview Statistics, National Center for Health Statistics, released December 2015, .
::1. " 无偿替代:2015年1月至6月全国健康访谈调查估计数的早期发布 " ,由Stephen J. Blumberg博士和国家卫生统计中心卫生访谈统计司Julian V. Luke撰写,2015年12月发表。

章节大纲

常规

套套套套套套

Sets ::套套套套套套

Example 1 ::例1

Subsets ::子设置

Complements ::补充

Union and Intersection ::工会和跨部门组织

Union and Intersection ::工会和跨部门组织

Example 2 ::例2

Venn Diagrams ::文文图

Example 3 ::例3

Example 4 ::例4

Feature: European Union ::特点:欧洲联盟

Coal, Steel, Trade and Peace ::煤炭、钢铁、贸易与和平

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

References ::参考参考资料

Sets
::套套套套套套

Example 1
::例1

Subsets
::子设置

Complements
::补充

Union and Intersection
::工会和跨部门组织

Union and Intersection
::工会和跨部门组织

Example 2
::例2

Venn Diagrams
::文文图

Example 3
::例3

Example 4
::例4

Feature: European Union
::特点:欧洲联盟

Coal, Steel, Trade and Peace
::煤炭、钢铁、贸易与和平

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX

References
::参考参考资料