Venn Diagrams and Set Notation
At GCSE, Venn diagrams come with their own short-hand: ξ, ∈, ∩, ∪ and ′. Once you can read and shade expressions like A′ ∩ B or (A ∪ B)′, you can fill in diagrams from clues (sometimes with algebra), work with three circles, and read probabilities straight off them.
GCSE subject content: enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams; calculate probabilities from them (Higher: conditional probabilities from Venn diagrams)Before you start
- The basics are on the KS3 page sample spaces and Venn diagrams: sorting numbers into circles, ξ, A ∩ B, A ∪ B and A′, and filling in a Venn diagram by starting in the middle. This page goes further.
- P(event) = outcomes you want ÷ all outcomes, and P(not A) = 1 − P(A). See mutually exclusive outcomes.
- For the algebra part: forming and solving a linear equation. See linear equations.
1 · Sets, elements and the symbols
A set is a collection of things, called elements (or members). We list them inside curly brackets { }, separated by commas, each one written once. The order doesn't matter: {1, 2, 3} and {3, 1, 2} are the same set.
| Symbol | Say it | Example |
|---|---|---|
| ξ | the universal set ("xi"): everything we are talking about | ξ = {1, 2, …, 15} |
| { } | the set containing… | A = {2, 3, 5, 7} |
| ∈ | is an element of | 5 ∈ A |
| ∉ | is not an element of | 4 ∉ A |
| n(A) | the number of elements in A | n(A) = 4 |
| ∅ or { } | the empty set: no elements | {even primes > 2} = ∅ |
Try it. ξ = {1, 2, …, 15}. Tap the numbers that belong in set A, then press Check. After that, decide whether each statement is true or false.
2 · Intersection, union and complement
Three operations make new sets from old ones:
| Notation | Say it | Means |
|---|---|---|
| A ∩ B | A intersection B | in A and in B (the overlap) |
| A ∪ B | A union B | in A or B or both (anywhere in the circles) |
| A′ | A complement, "A dash", "not A" | not in A, including outside both circles |
Here ξ = {1, 2, …, 20} has been sorted for you. Tap regions to shade them (tap again to unshade). The panel names the set you've shaded in set notation, lists its elements and gives the probability that a number picked at random from ξ is in it.
Memory tricks. ∩ looks like a bridge joining two banks: you need both ("and"). ∪ is a cup that holds anything poured in ("or"). The dash ′ means "everything else in ξ", so A′ includes the region outside both circles.
3 · Building combinations step by step
Exam questions combine the operations: A′ ∩ B, (A ∪ B)′, A ∪ B′. Do them in order, like BIDMAS: brackets first, then dashes on single letters, then ∩ or ∪. For ∩, keep only where both shadings overlap. For ∪, keep everything shaded by either. Pick an expression and press Next step.
A trap worth knowing. (A ∪ B)′ is "outside both circles", which is the same as A′ ∩ B′ (not A and not B). But A′ ∪ B′ is everything except the overlap: it's (A ∩ B)′. Moving the dash inside a bracket swaps ∩ and ∪.
4 · The reverse game: name the shading
Now the other way round: a region is shaded, you pick the expression. If you pick the wrong one, you'll be told which regions your choice would shade, so you can see the difference.
5 · Filling in a Venn diagram from clues
When each region holds a count, the KS3 rule was "start in the middle". At GCSE the clues are often given in set notation, and the middle isn't handed to you. The key fact: n(A) + n(B) counts the overlap twice, so
n(A ∩ B) = n(A) + n(B) − n(A ∪ B)
and n(A ∪ B) = n(ξ) − n((A ∪ B)′). Fill in all four regions and press Check, or press Show me step by step.
6 · Venn diagrams with probabilities
A Venn diagram can hold probabilities instead of counts. Then everything in ξ adds up to 1, so a missing region is 1 minus the rest. Probabilities of sets are found by adding regions, just like counts. Set the three regions inside the circles; the outside fills itself in.
Notice the addition rule hiding in there: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Adding P(A) and P(B) counts the overlap twice, so take it off once. The conditional probability P(A | B) (Higher) means "A given B": only look inside B, so divide the overlap by P(B). There's a whole page on it: conditional probability.
7 · Venn diagrams with algebra Higher
Sometimes there aren't enough numbers to start in the middle. Then call the unknown region x, write every other region in terms of x, and use the fact that all the regions add up to n(ξ) to form an equation. Drag the slider to try values of x and watch whether the total balances; then solve it properly.
8 · Three-set Venn diagrams Higher
Three circles A, B and C make 8 regions: the very middle A ∩ B ∩ C, three "two-only" regions (like A ∩ B ∩ C′, in A and B but not C), three "one-only" regions, and the outside (A ∪ B ∪ C)′. Every element goes in exactly one. Tap regions, or pick an expression, and the elements of ξ = {1, 2, …, 30} light up.
To fill a three-set diagram from counts, the golden rule still holds: start in the very middle (all three), then the three "two-only" regions (subtract the middle from each pair's total), then the "one-only" regions (subtract everything else inside that circle), and the outside last. See the last worked example.
Common mistakes and exam traps
- Mixing up ∩ and ∪. ∩ = and = overlap only; ∪ = or = anywhere in either circle.
- Forgetting the outside. A′ includes the region outside both circles.
- Writing an element twice when listing A ∪ B. Each element once.
- Putting n(A) straight into circle A's outer part. n(A) includes the overlap: A only = n(A) − n(A ∩ B).
- (A ∪ B)′ is not A′ ∪ B′. Do the bracket first.
- Regions not adding up. Always check the total equals n(ξ) (or 1 for probabilities).
Worked examples
ξ = {1, 2, …, 12}, A = {factors of 12}, B = {even numbers}. List A ∩ B, A ∪ B and A′ ∩ B, and find P(A′ ∩ B).
Describe the regions (A ∪ B)′ and A′ ∪ B′ and explain why they are different.
n(ξ) = 50, n(A) = 20, n(B) = 25 and n((A ∪ B)′) = 12. Complete the Venn diagram and find P(A′ ∩ B).
In a Venn diagram, P(A only) = 0.35, P(A ∩ B) = 0.15 and P((A ∪ B)′) = 0.2. Find P(B only), P(A ∪ B), P(B′) and (Higher) P(A | B).
Exam-style Higher: 60 students. 35 play a sport (S), 28 play an instrument (I) and 7 do neither. Find how many do both.
Exam-style Higher: 100 students. 40 study French, 35 German, 30 Spanish. 12 study French and German, 9 French and Spanish, 8 German and Spanish, and 5 study all three. Find P(a student picked at random studies exactly one language).
Quick check
What you've learned
- ξ is the universal set; { } lists elements once each; ∈ / ∉ mean "is / is not an element of"; n(A) counts the elements.
- A ∩ B = and (overlap); A ∪ B = or (either or both); A′ = not A (including outside the circles).
- Build combinations in order: brackets, then dashes, then ∩ (overlap of shadings) or ∪ (all shadings). (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
- From clues: n(A ∩ B) = n(A) + n(B) − n(A ∪ B). Check the regions add to n(ξ).
- With probabilities, all regions add to 1; P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Higher: use x for an unknown region and form an equation from the total; three-set diagrams have 8 regions and you fill them from the middle out.