A set is just a tidy collection of things — the days of the week, the factors of 12, the pupils who study French. Mathematicians write sets with a special shorthand so that overlaps and combinations can be described precisely, and Venn diagrams turn that shorthand into a picture you can count from.
1. The core notation
| Symbol | Means | Example |
|---|---|---|
| ℰ | universal set (everything) | all whole numbers 1–10 |
| ∈ / ∉ | is / is not a member of | 3 ∈ {1,2,3}; 5 ∉ {1,2,3} |
| ∩ | intersection (in BOTH) | {1,2,3} ∩ {2,3,4} = {2,3} |
| ∪ | union (in EITHER) | {1,2} ∪ {2,3} = {1,2,3} |
| A′ | complement (NOT in A) | everything in ℰ outside A |
| ⊆ | subset of | {1,2} ⊆ {1,2,3} |
| ∅ | empty set | { } — nothing in it |
| n(A) | number of members of A | n({1,2,3}) = 3 |
Memory aids: ∪ (union) looks like a cup that holds everything from both sets; ∩ (intersection) is the small overlap where they meet.
2. Two- and three-set Venn diagrams
- Each region is a distinct combination of the sets. With three sets there are eight regions (including "outside all three").
- Fill from the middle out: start with the centre A∩B∩C, then the pairwise overlaps, then the single regions, then the outside.
3. Solving with numbers in regions
Notice n(M ∪ S) = n(M) + n(S) − n(M ∩ S) = 18 + 15 − 8 = 25 — the overlap is subtracted once because it was counted twice.
Exam technique: the command word Complete the Venn diagram means fill in the centre overlap first, then work outwards — this stops you double-counting. For n(A ∪ B) add the circle totals and subtract the intersection once.