Set language and notation

Key learnings

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  Learn the core set symbols

  The symbols you must read and write fluently: $\in$ (is an element of), $\notin$ (is not), $\subseteq$ (subset of), $\subset$ (proper subset), $A \cup B$ (union — in either or both), $A \cap B$ (intersection — in both), $A'$ (complement — everything not in $A$), $\xi$ (the universal set), $\varnothing$ (the empty set), and $n(A)$ (the number of elements in $A$).

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  Union vs intersection: "or" vs "and"

  $A \cup B$ ("union") is everything in $A$ or $B$ (or both); $A \cap B$ ("intersection") is only what is in $A$ and $B$. Picture a Venn diagram: union shades both circles, intersection shades just the overlap.

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  Counting a union: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

  When you add $n(A)$ (n(A) is the number of elements in $A$) and $n(B)$ you count the overlap twice, so subtract $n(A \cap B)$ once. This inclusion–exclusion rule lets you count a union without listing every element.