The sets of numbers
Each set of numbers sits inside a bigger one. The diagram shows how they nest: the natural numbers are a subset of the whole numbers, the whole numbers a subset of the integers, and so on.
What each set means - notation, words and examples
A note on notation: the irrationals are sometimes written ℝ∖ℚ ("real but not rational"). That slash is set difference, not division.
Set notation at a glance
Every symbol you meet in GCSE set notation, with the shaded region it stands for. A is the blue circle, B the red circle, and the rectangle is the universal set ξ.
Shading sets with diagonals
The reliable way to shade a Venn diagram is to shade each set with its own diagonal lines, then read off the answer. Shade set A with lines leaning one way and set B with lines leaning the other.
Union (A ∪ B = "or") is everywhere with any shading at all. Intersection (A ∩ B = "and") is only the cross-hatched part, where an element is in A and B at once.
Sets In Context
Given
- ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- A = even numbers = {2, 4, 6, 8, 10}
- B = multiples of 3 = {3, 6, 9}
Find these sets
Work each one out, then click to reveal the answer.
A ∩ B = ?
A ∪ B = ?
(A ∪ B)′ = ?
Conditional probability
"Given" throws away part of the sample space - the condition (highlighted yellow) becomes the denominator, and the answer (highlighted green) is the numerator.
Venn Diagrams
40 students study French (A), Spanish (B), both or neither. Find P(Spanish | French).
Show highlighting & answer
"Given French" keeps only the yellow set; the part also doing Spanish is green (outlined yellow, so you can see it is still part of the French total).
Your turn: find P(French | Spanish).
Show answer
Tree Diagrams
A bag holds 3 red and 2 blue balls. Two are taken without replacement. Find P(different colours | at least one red).
Show highlighting & answer
The yellow branches meet the condition; the green ones (drawn on top of yellow) are the answer.
Your turn: find P(both red | at least one red).
Show answer
Two Way Tables
50 students were asked if they play a school sport. Find P(sport | girl).
Show highlighting & answer
"Given girl" uses only the Girls row; the answer cell is green (outlined yellow, so you can see it sits inside the Girls row).
| Sport | No sport | Total | |
|---|---|---|---|
| Boys | 18 | 7 | 25 |
| Girls | 15 | 10 | 25 |
| Total | 33 | 17 | 50 |
Your turn: find P(girl | sport).
Show answer
Number Sets activity
Drag each number into the smallest ring it belongs to. The rings are nested, so every natural number is also an integer, every integer is rational, and so on. (Tap a number then tap a ring on a touchscreen.)
Shading Sets activity
A worksheet of Venn diagrams to shade. Click the regions that belong to each set, then press Check. Work down through the five levels - they get harder.
Conditional Probability activity
Each question is a conditional probability, P(A | B). Sketch the fraction in two colours: click a region/branch/cell once to shade it yellow (the denominator - the "given"), twice for green (the numerator - the answer part), and a third time to clear. The probability is then green out of green + yellow.
Exam-style questions
One harder question on each representation. Work it through, then reveal the full solution.