Mathematics 0580 · Syllabus tracker
Sets
Set notation, listing elements, complements, Venn diagrams with two or three sets, shading and word problems.
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Quick revision
Sets on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Set language and symbols, Venn diagrams with two or three sets, shading, and word problems.
1Set notation
- Learn the symbols — questions use them without explanation.
| Symbol | Meaning |
|---|---|
| ξ | the universal set (everything) |
| x ∈ A | x is in A |
| x ∉ A | x is not in A |
| A′ | everything not in A |
| A ∪ B | in A or B or both (union) |
| A ∩ B | in both A and B (intersection) |
| ∅ | the empty set |
| A ⊆ B | A is a subset of B |
| n(A) | the number of elements in A |
| {x : x > 3} | the set of x such that x > 3 |
Worked example
A = {2, 4, 6, 8, 10}, B = {3, 6, 9}. Find A ∩ B and n(A ∪ B).
- Both: 6.
- A ∪ B = {2, 3, 4, 6, 8, 9, 10}.
A ∩ B = {6}; n(A ∪ B) = 7
2Venn diagrams and shading
- Two sets make four regions: A only, both, B only, neither.
- A ∩ B′ = in A but not B. (A ∪ B)′ = outside both. A′ ∪ B = everything except 'A only'.
- For three sets, there are eight regions; fill from the centre outwards.
| Region | Notation |
|---|---|
| A only | A ∩ B′ |
| B only | A′ ∩ B |
| Both | A ∩ B |
| Neither | (A ∪ B)′ or A′ ∩ B′ |
| Exactly one | (A ∩ B′) ∪ (A′ ∩ B) |
Worked example
Describe the region 'in B but not in A or C' using set notation.
- In B, not in A, not in C.
B ∩ A′ ∩ C′
3Venn diagram problems
- Start with the overlap, then subtract it from each set's total.
- Remember the region outside the circles.
- With an unknown, write each region in terms of x and set the total equal to n(ξ).
- Probability from a Venn diagram = number in the region ÷ the total (or ÷ the given set for 'given that').
Formulas
Worked example
In a group of 50, 32 like tea, 25 like coffee and 6 like neither. How many like only tea?
- Let x like both: (32 − x) + x + (25 − x) + 6 = 50.
- 63 − x = 50, x = 13.
- Only tea = 32 − 13.
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✗Common mistakes
| Don’t write | Write instead |
|---|---|
| n(A ∪ B) = n(A) + n(B) | Subtract the overlap |
| Putting 32 in the 'tea only' region | 32 is all of tea; tea only = 32 − both |
| Forgetting the region outside the circles | It is part of ξ — include it in the total |
| Listing 1 as a prime in a set | 1 is not prime |
★In the exam
- Draw the Venn diagram even when not asked — it organises the information.
- Check that all the regions add up to n(ξ).
- Shading questions: shade carefully and clearly; mark the region boundaries.