Sets
Introduced| English | Meaning |
|---|---|
| set/set/ | A collection of distinct elements. |
| elements/ˈelɪmənts/ | Individual members of a set. |
| universal set/ˌjuːnɪˈvɜːsl set/ | The set of all elements being considered in a particular problem. |
| empty set/ˈempti set/ | The set with no members, written ∅ or {}. |
| intersection/ˌɪntəˈsekʃn/ | The set of elements that belong to all of the sets being compared. |
| union/ˈjuːnɪən/ | The set of elements that belong to at least one of the sets being combined. |
| complement/ˈkɒmplɪmənt/ | The set of elements in a specified universal set that do not belong to the given set. |
| Venn diagram/ven ˈdaɪəɡræm/ | A diagram using overlapping regions to represent sets and their relationships. |
The class of 2024
- A school has 30 students. 18 study French, 12 study Spanish, and 5 study both.
- How many study neither? You can't just add — you need sets 集合.
- Sets are the language of grouping, and they power everything from databases to probability.
What is a set?
- A set is a collection of distinct objects (the elements 元素 or members).
- Written with curly braces: $A = \{1, 2, 3, 4, 5\}$.
- $3 \in A$ means "$3$ is an element of $A$"; $7 \notin A$ means "$7$ is not".
- The universal set 全集 $\xi$ contains everything under discussion. The empty set 空集 $\emptyset$ has no elements.
$A\cap B=\{6\}$ is the only number in both circles; $A\cup B$ is everything inside either circle
Set operations on a Venn diagram
A Venn diagram shows two sets. Slide through the operations to see which region union, intersection and complement shade in.
The empty set ∅ is a subset of every set.
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The empty set has no elements, so it is trivially contained in every set.
Intersection 交集 and union 并集
- Intersection $A \cap B$: elements in both $A$ and $B$.
- Union $A \cup B$: elements in $A$ or $B$ (or both).
- Complement 补集 $A'$: everything in the universal set that is not in $A$.
$A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$. Then $A \cap B = \{3, 4\}$ and $A \cup B = \{1, 2, 3, 4, 5, 6\}$.
A = {2,4,6,8,10} and B = {3,6,9}. How many elements are in A ∩ B?
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Only 6 is in both sets, so A ∩ B = {6}, which has 1 element.
For the same A and B, how many elements are in A ∪ B?
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A ∪ B = {2,3,4,6,8,9,10}, which has 7 elements.
The symbol ∩ means:
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∩ is intersection (elements in both A and B); ∪ is union.
Venn diagrams 维恩图
- A Venn diagram draws sets as overlapping circles inside a rectangle (the universal set).
- Overlapping region = intersection; everything inside a circle = that set's members.
Venn diagrams show how sets relate — here, every natural number is also an integer and a rational.
Don't double-count. When finding $n(A \cup B)$, use $n(A) + n(B) - n(A \cap B)$. The intersection is counted once in each set, so subtract it once.
Worked example
- 30 students, $F =$ French (18), $S =$ Spanish (12), $F \cap S = 5$.
- $n(F \cup S) = 18 + 12 - 5 = 25$.
- Neither $= 30 - 25 = 5$ students study neither language.
n(A) = 5, n(B) = 3, n(A ∩ B) = 1. Find n(A ∪ B).
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n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 5 + 3 − 1 = 7.
In a class of 30, 18 study French, 12 study Spanish, and 5 study both. How many study neither?
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n(F ∪ S) = 18 + 12 − 5 = 25. Neither = 30 − 25 = 5.
Three sets (Extended)
- In a three-set Venn diagram, fill the triple overlap first, then the parts belonging to exactly two sets, then the single-set regions. If $n(A\cap B)=7$ includes 2 in all three, the AB-only region is $7-2=5$.
- For $n(A)=12,n(B)=10,n(C)=9$, pair intersections AB=4, AC=3, BC=2 and triple=1, the union is $n(A\cup B\cup C)=12+10+9-4-3-2+1=23$. The triple is added back because the three pair subtractions removed it too often.
Set sizes are 12,10,9; pair intersections 4,3,2; triple intersection 1. Find the union size.
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Add singles, subtract all pairs, add triple: 31−9+1 = 23.
You've got it
- $A \cap B$ = intersection (both); $A \cup B$ = union (either or both); $A'$ = complement
- $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ — subtract the overlap
- Venn diagrams show set relationships visually; the rectangle is the universal set
- $\emptyset$ = empty set; $\xi$ = universal set
Write from an intersection of 2
- A is {1, 2, 3, 4, 5} and B is {4, 5, 6, 7}.
- Count the elements in both sets.
A is the set 1, 2, 3, 4, 5 and B is the set 4, 5, 6, 7. How many elements are in A intersection B?
Only 4 and 5 belong to both sets.
How many elements are in the intersection?
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Two numbers are shared. Other options — 1 (the lesson intersection); 7 (the union).