Sets

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

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 $A$.

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.

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.

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.

You've got it

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.