Set notation in O-Level E Maths uses: n(A) for the number of elements in set A, A ∩ B for intersection (elements in both A and B), A ∪ B for union (elements in A or B or both), A' for the complement (everything not in A), and A ⊂ B to mean A is a subset of B. The key skill is translating between set notation, a Venn diagram's shaded regions, and the actual count of elements — most exam questions test all three at once.
Set notation questions aren't computationally difficult — there's rarely more than basic addition and subtraction involved. What trips students up is the translation step: converting between the symbol on the page, the shaded region on a Venn diagram, and what that region actually represents in a word problem. Get comfortable with that translation, and the rest is arithmetic.
n(A)
The number of elements in set A. If A = {2, 4, 6, 8}, then n(A) = 4. This counts elements, it does not list them.
A ∩ B
Intersection — elements in both A and B. On a Venn diagram, this is the overlapping region (shaded).
A ∪ B
Union — elements in A or B or both. On a Venn diagram, this is both circles shaded entirely.
A'
Complement — everything in the universal set ξ that is not in A. On a Venn diagram, this is everywhere outside circle A but inside the rectangle.
A ⊂ B
A is a subset of B — every element of A is also in B. On a Venn diagram, circle A is drawn entirely inside circle B.
a ∈ A
"a" is an element of set A — a is a single member inside the set. On a Venn diagram, this is a single point plotted inside circle A.
In a class of 40 students, 25 study French, 18 study Spanish, and 8 study neither language. Find the number of students who study both French and Spanish.
Solution
n(F ∪ S) = 40 − 8 = 32n(F ∪ S) = n(F) + n(S) − n(F ∩ S)32 = 25 + 18 − n(F ∩ S)32 = 43 − n(F ∩ S)n(F ∩ S) = 11Memorise this formula — it appears in almost every 2-set word problem: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). It exists because elements in the overlap get counted twice if you simply add n(A) + n(B), so the overlap must be subtracted once to correct for it.
For any 2-set Venn diagram question, always fill in the regions starting with the intersection and working outward — this order avoids double-counting.
Continuing the example — filling in all 4 regions
Common mistake
Students often place 25 directly into the F circle and 18 directly into the S circle without subtracting the overlap first — this double-counts the 11 students in both. Always calculate and place the intersection value first, then subtract it from each individual total before filling in the "only" regions.
Some questions skip the word problem and just show a Venn diagram with a shaded region, asking you to identify the correct set notation for it.
| Shaded region description | Set notation |
|---|---|
| Only the overlap between A and B | A ∩ B |
| Everything in either circle | A ∪ B |
| Inside A only, not in B | A ∩ B' |
| Inside B only, not in A | A' ∩ B |
| Outside both circles entirely | (A ∪ B)' or A' ∩ B' |
| Everywhere except the overlap | (A ∩ B)' |
"A only" always means "in A, but not in B" — written as A ∩ B'. Students often just write "A" for this region, but that's incorrect since set A on its own includes the overlap too. The prime symbol on B is what excludes the overlapping part.
The same inside-out filling strategy extends to three overlapping sets — just start with the very centre (all three sets) and work outward through each pairwise overlap before the "only" regions.
What's the difference between n(A) and A itself?
A refers to the set itself — its actual elements, e.g. A = {2, 4, 6}. n(A) refers to the number of elements in that set — a single number, e.g. n(A) = 3. Writing n(A) = {2, 4, 6} is a common notation error; n(A) must always be a number, never a list.
What is the empty set, and how is it written?
The empty set contains no elements at all, written as ∅ or { }. If two sets don't overlap at all, their intersection is the empty set: A ∩ B = ∅. This is a common answer when a Venn diagram question describes mutually exclusive categories, the same idea tested directly in probability of an event questions.
Does the order matter in A ∩ B vs B ∩ A?
No — both intersection and union are commutative, meaning A ∩ B = B ∩ A and A ∪ B = B ∪ A. The overlapping region on a Venn diagram is the same regardless of which set you name first.
How do I know if a question wants a Venn diagram drawn, or just the calculation?
Read the command word carefully. "Draw a Venn diagram to represent..." requires an actual diagram with regions labelled or shaded. "Find n(A ∩ B)" or "how many students study both..." usually only requires the calculation — though sketching a quick diagram as working is still good practice, since it helps you avoid the double-counting mistake.
— Mr Gan Math Tuition
Mr. Gan teaches a diagram-first method for sets and Venn diagrams, so notation questions become quick marks instead of traps.
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