Sets, relations and functions · 集合、关系与函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| set/set/ | 集合 | jí hé |
| union/ˈjuːnɪən/ | 并集 | bìng jí |
| intersection/ˌɪntəˈsekʃn/ | 交集 | jiāo jí |
| set complement/set ˈkɒmplɪmənt/ | 补集 | bǔ jí |
| subset/ˈsʌbset/ | 子集 | zi jí |
| Venn diagram/ven ˈdaɪəɡræm/ | 韦恩图 | wéi ēn tú |
| inclusion-exclusion principle/ɪnˈkluːʒn eksˈkluːʒn ˈprɪnsɪpl/ | 容斥原理 | róng chì yuán lǐ |
| relation/rɪˈleɪʃn/ | 关系 | guān xì |
| function/ˈfʌŋkʃn/ | 函数 | hán shù |
Count a shared student once
- A class of 30 includes 18 French students, 15 German students and 7 taking both. Adding the subject totals counts the shared seven twice.
- A set 集合 contains distinct objects. Order and repeated entries do not change its membership or number of elements.
将共享的学生计数一次
- 一个30人的班级中有18名法语学生、15名德语学生和7名同时修读两门课的学生。若直接将科目人数相加，会将这7名共享学生重复计算两次。
- 集合 包含互不相同的元素。元素的排列顺序和重复记录不会改变其成员身份或元素数量。
State the universe for a complement
- The union 并集 contains elements in either set; the intersection 交集 contains elements in both.
- A set complement 补集 contains elements of the stated universe outside the set. A subset 子集 contains only elements belonging to its parent set.
明确补集的全域
- 并集 包含属于任一集合的元素；交集 包含同时属于两个集合的元素。
- 补集 包含全域中不属于该集合的元素。子集 仅包含其父集合中的元素。
Match each set operation to what it contains. · 将每个集合运算与其包含内容匹配。
Union contains either-set members, intersection contains both-set members, and a complement contains only members of the stated universe outside the set. · 并集包含任一集合的成员，交集包含两集合共有的成员，补集仅包含声明论域中不属于该集合的成员。
Partition the count into disjoint regions
- A Venn diagram 韦恩图 can display the overlap and the only/neither regions. Check that those regions sum to the supplied universe total.
- The inclusion-exclusion principle 容斥原理 gives $|A\cup B|=|A|+|B|-|A\cap B|$, subtracting the twice-counted overlap once.
将计数划分为不相交的区域
- 韦恩图 可以显示重叠区域以及“仅属于”或“两者皆非”的区域。需验证这些区域的总和是否等于给定的全域总数。
- 容斥原理 给出 $|A\cup B|=|A|+|B|-|A\cap B|$，即减去被重复计算的交集部分一次。
Two sets and their overlap · 两个集合及其重叠部分
The union counts the overlap once, which is what the minus sign is for. · 并集只计重叠一次，这就是减号的作用。
The supplied class. French-only is 11, both 7, German-only 8 and neither 4. The union is 26 and the disjoint total is $11+7+8+4=30$. Neither means outside both subjects but still inside this class universe.
给定的班级数据。 仅法语11人，两科均修7人，仅德语8人，两科均未修4人。并集为26人，不相交部分的总计为 $11+7+8+4=30$。“两者皆非”指在两科之外但仍在该班级全域之内的学生。
In a class of 30, 18 study French, 15 study German and 7 study both. How many study neither? · 一个班级30人，18人选法语，15人选德语，7人选两门。多少人没选？
18 + 15 − 7 = 26 study at least one, so 30 − 26 = 4 study neither. · 18 + 15 − 7 = 26 至少选一科，所以 30 − 26 = 4 两门都没选。
In the same class, how many study French only? · 同班中，多少人只选法语？
18 − 7 = 11. Filling the Venn diagram from the middle outwards never double-counts. · 18 − 7 = 11。从中间向外填充韦恩图永不重复计数。
Each domain input needs exactly one output
- A relation 关系 pairs elements. A function 函数 assigns exactly one output to every input in its stated domain; different inputs may share an output.
- For $f(x)=x^2$ on negative and positive inputs, the same output can have two inputs. Its inverse relation is then not a function on that range without a suitable restriction.
每个定义域输入必须恰好对应一个输出
- 关系 将元素配对。函数 将其给定定义域内的每个输入分配恰好一个输出；不同输入可能共享同一输出。
- 对于负数和正数输入上的 $f(x)=x^2$，同一输出可对应两个输入。若无适当的限制条件，其逆关系在该值域上便不是函数。
What makes a relation a function? · 什么使关系成为函数？
Every input in the stated domain must have exactly one output. Different inputs may share an output. · 声明定义域内的每个输入必须有且仅有一个输出。不同输入可能共享同一输出。
Separate membership from counting. In sheet 4.1, list set elements before reporting their counts, and identify the stated domain before judging whether a relation is a function or has an inverse function.
区分成员身份与计数。 在练习纸4.1中，列出集合元素后再报告其数量，并在判断关系是否为函数或是否存在逆函数之前，先明确给定的定义域。
What does |A| mean in set notation? · 集合记法中 |A| 表示什么？
In set notation |A| counts the distinct elements of A. · 在集合记号中 |A| 统计 A 的不同元素个数。
In set notation $|A|$ is the number of elements of A. A complement without its universe is incomplete: an element outside the universe is not part of that complement.
在集合符号 $|A|$ 中表示A的元素个数。未指明全域的补集是不完整的：全域之外的元素不属于该补集。