Sets, relations and functions
Introduced| English | Meaning |
|---|---|
| set/set/ | A collection of distinct elements. |
| union/ˈjuːnɪən/ | The set of elements that belong to at least one of the sets being combined. |
| intersection/ˌɪntəˈsekʃn/ | The set of elements that belong to all of the sets being compared. |
| set complement/set ˈkɒmplɪmənt/ | The elements of a specified universal set that do not belong to the given set. |
| subset/ˈsʌbset/ | A set all of whose elements also belong to another specified set. |
| Venn diagram/ven ˈdaɪəɡræm/ | A diagram using overlapping regions to represent sets and their relationships. |
| inclusion-exclusion principle/ɪnˈkluːʒn eksˈkluːʒn ˈprɪnsɪpl/ | A method of counting a union by adding individual set sizes, subtracting pairwise overlaps, and continuing with alternating signs for larger intersections. |
| relation/rɪˈleɪʃn/ | A function or has an inverse function. |
| function/ˈfʌŋkʃn/ | A mapping that assigns exactly one output to each allowed input. |
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.
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.
Match each set operation to what it contains.
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Union contains either-set members, intersection contains both-set members, and a complement contains only members of the stated universe outside the set.
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.
Two sets and their overlap
The union counts the overlap once, which is what the minus sign is for.
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.
In a class of 30, 18 study French, 15 study German and 7 study both. How many study neither?
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18 + 15 − 7 = 26 study at least one, so 30 − 26 = 4 study neither.
In the same class, how many study French only?
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18 − 7 = 11. Filling the Venn diagram from the middle outwards never double-counts.
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.
What makes a relation a function?
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Every input in the stated domain must have exactly one output. Different inputs may share an output.
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.
What does |A| mean in set notation?
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In set notation |A| counts the distinct elements of A.
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.