Set Notation and Venn Diagrams

This topic is mostly vocabulary. Once the symbols are secure, the questions are straightforward — and when they aren’t secure, every mark in the question goes. Union and intersection being confused with each other was the most frequent error recorded in real lessons.

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1. The symbols

Symbol	Name	Meaning
∪	Union	in A OR B or both — everything in either
∩	Intersection	in A AND B — only what is in both
A′	Complement	everything NOT in A
ξ	Universal set	everything under consideration
∈	is an element of	3 ∈ A means 3 is in A
∉	is not an element of
⊂	is a subset of	every element of A is also in B
n(A)	number of elements	how many are in A
∅ or { }	empty set	no elements

Union is the bigger one; intersection is the overlap. Two memory hooks that work: ∪ looks like a cup that holds everything, and ∩ is the cap sitting over just the overlap. Alternatively, ∪ for Union contains the “or”; ∩ contains the “and”.

Intersection means what the sets have in COMMON. One student read it as “not together” — the opposite of the truth. If the sets share nothing, A ∩ B = ∅.

A′ means everything outside A, within the universal set — not “the rest of some other set”.

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2. Listing sets

If ξ = {1,2,3,4,5,6,7,8,9,10}, A = {2,4,6,8,10}, B = {1,2,3,4,5}:

• A ∪ B = {1,2,3,4,5,6,8,10}
• A ∩ B = {2,4}
• A′ = {1,3,5,7,9}
• n(A) = 5
• n(A ∩ B) = 2

Never repeat an element in a union. 2 and 4 are in both sets but appear once each in A ∪ B. Listing them twice was a documented error.

Write elements in increasing order inside the braces — tutors flagged this repeatedly as an easy presentation mark.

n(A) is a NUMBER; A is a LIST. If the question asks for n(A ∩ B), write 2, not {2,4}. Answering with the wrong type is a common and entirely avoidable loss.

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3. Reading a Venn diagram

Two overlapping circles inside a rectangle (the universal set) create four regions:

1. A only — in A, not B
2. The overlap — A ∩ B
3. B only — in B, not A
4. Outside both — (A ∪ B)′

“Only” changes the answer. “How many study French only” wants the region excluding the overlap. “How many study French” wants the whole circle, overlap included. If the word “only” is absent, take the whole set — this distinction was flagged by tutors and got wrong by students more than almost anything else here.

Worked reading. If A only = 7, overlap = 3, B only = 5, outside = 4:

• n(A) = 7 + 3 = 10
• n(B) = 3 + 5 = 8
• n(A ∩ B) = 3
• n(A ∪ B) = 7 + 3 + 5 = 15
• n(A′) = 5 + 4 = 9
• n(ξ) = 7 + 3 + 5 + 4 = 19

Add every region inside the shape. n(A) includes the overlap. The most common slip is reading only the “A only” region.

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4. Filling in a Venn diagram

Always start with the intersection, then work outwards.

Example: 30 students; 18 play tennis, 15 play hockey, 8 play both.

1. Overlap = 8
2. Tennis only = 18 − 8 = 10
3. Hockey only = 15 − 8 = 7
4. Neither = 30 − (10 + 8 + 7) = 5

Subtract the overlap from each total. Writing 18 and 15 in the “only” regions double-counts the 8 and makes the total exceed 30 — the standard error. Check your regions sum to n(ξ); it catches this immediately.

With algebra

If the overlap is unknown, call it x, write each region in terms of x, set the total equal to n(ξ), and solve.

Example: 40 people; 25 like tea, 20 like coffee, 3 like neither. How many like both?

• Let both = x
• Tea only = 25 − x, coffee only = 20 − x
• (25 − x) + x + (20 − x) + 3 = 40
• 48 − x = 40 → x = 8

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5. Shading regions

You may be asked to shade a region for a given expression, or to write the notation for a shaded region.

Method — work from the inside of the notation outwards:

Expression	Shade
A ∪ B	both circles entirely
A ∩ B	the overlap only
A′	everything outside A, including B-only and the outside region
(A ∪ B)′	only the region outside both
A ∩ B′	the part of A not in B — “A only”
A ∪ B′	all of A, plus everything outside B

A dash applies to whatever it is attached to. In A ∩ B′ the dash is on B alone, so you want the part of A outside B. In (A ∪ B)′ the dash applies to the whole bracket.

Shade only the region asked for. Shading whole circles when only the overlap was wanted — and shading the middle when it should be excluded — were both recorded errors.

A reliable check: invent small numbers for each region, or pick a sample element, and test whether it belongs in the expression. If an element in “B only” does not satisfy A ∩ B′, that region should not be shaded.

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6. Three-set problems

With three circles there are eight regions. The method is the same but the order matters:

Start with the centre — the region in all three — then the pairwise overlaps, then the single regions, then the outside.

Each pairwise overlap must have the centre subtracted from it before you write it in.

For counting, the formula is:

n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(A∩C) − n(B∩C) + n(A∩B∩C)

Filling the diagram from the middle outwards is usually safer than the formula, and it also answers the “only” questions that normally follow.

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7. Sets and probability

Venn diagram questions frequently end with a probability.

P(event) = (number in that region) ÷ n(ξ)

Using the earlier example (n(ξ) = 19, overlap = 3): P(a student is in both) = 3/19.

The denominator is the total in the universal set, not the total in one circle — unless the question restricts it (“given that the student studies French…“).

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8. Mistakes that cost marks

Confusing ∪ with ∩ — the dominant error in this topic.

Reading intersection as “not in common”.

Repeating an element in a union.

Giving a list when n(…) asked for a number, or vice versa.

Ignoring the word “only” — or assuming it when it isn’t there.

Reading n(A) as the “A only” region.

Not subtracting the overlap when filling in a diagram.

Forgetting the “neither” region outside the circles.

Misapplying the dash in expressions like A ∩ B′.

Shading whole circles instead of the specific region.

Not listing elements in increasing order.

Using the wrong denominator in a probability.

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Frequently asked questions

What does ∪ mean? Union — everything in either set (or both).

What does ∩ mean? Intersection — only what is in both sets.

How do I remember the difference? ∪ is a cup holding everything; ∩ is a cap covering just the overlap.

What does A′ mean? The complement — everything in the universal set that is not in A.

What does n(A) mean? The number of elements in A — a number, not a list.

What is the universal set? ξ — everything being considered, drawn as the rectangle.

How do I fill in a Venn diagram? Start with the intersection, subtract it from each set total, then find the “neither” region so the total matches n(ξ).

What’s the difference between “A” and “A only”? A is the whole circle including the overlap; A only excludes it.

How do I shade A ∩ B′? The part of A that is outside B — the “A only” region.

How do I find a probability from a Venn diagram? Divide the number in the region by the total number in the universal set.

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Quick revision checklist

• I know all the symbols: ∪, ∩, ′, ξ, ∈, ⊂, n(), ∅
• I can tell union from intersection without hesitating
• I list sets without repeats, in increasing order
• I know n(A) is a number and A is a list
• I can read all four regions of a two-set diagram
• I know n(A) includes the overlap
• I watch for the word only
• I fill diagrams starting from the intersection
• I check my regions total n(ξ)
• I can use algebra for an unknown overlap
• I can shade any expression, applying the dash correctly
• I can handle three-set diagrams from the centre outwards
• I can find a probability using n(ξ) as the denominator

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These notes cover set notation and Venn diagrams in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.