Sort the numbers into the right set
Drag each number into the smallest ring it belongs to (or tap a number then tap a ring). The rings are nested, so every natural number is also an integer, every integer is rational, and so on — but a number lives in the innermost ring it fits. Watch out: √9 is really 3, and 22/7 is a fraction, not π!
The sets of numbers
Each set of numbers sits inside a bigger one. The diagram shows how they nest: counting numbers inside whole numbers inside integers inside rationals — and rationals together with irrationals make up all the real numbers.
The real numbers split into rationals (which can be written as a fraction) and irrationals (which cannot).
What each set means — notation, words and examples
A note on notation: the irrational numbers are sometimes written ℝ∖ℚ ("real but not rational"). The slash there is set difference, not division — it means "take the rationals away from the reals".
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 — left lines, right lines, or both.
Intersection (A ∩ B = "and") is only the cross-hatched part — where the two sets of diagonals cross over, so an element is in A and B at once.
The universal set is ξ = {1,2,3,4,5,6,7,8,9,10}. Let A be the even numbers and B the multiples of 3. Here is every number placed in the diagram:
- A = {2, 4, 6, 8, 10} — the even numbers (left circle).
- B = {3, 6, 9} — the multiples of 3 (right circle).
- A ∩ B = the overlap = even and a multiple of 3 = {6}.
- A ∪ B = everything in either circle = {2, 3, 4, 6, 8, 9, 10}.
- (A ∪ B)′ = outside both circles = {1, 5, 7}.
Now try these — shade it in your head, then check
Shade A ∩ B
Shade A ∪ B
Shade A ∪ B′
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.
Practice levels — easy to hard (like Transum)
Click a level to load it into the diagram above. Each one is a little harder than the last.
Conditional probability
"Given" throws away part of the sample space — the condition becomes the new denominator. The same idea reads off a Venn diagram, a tree diagram and a two-way table.
Venn Overlaid sets
40 students study French, Spanish, both or neither. The two sets are overlaid: French in blue, Spanish in amber, so the overlap shows both colours.
Find P(Spanish | French). "Given French" keeps only the blue students.
The blue set has 12 + 8 = 20 (the denominator); the part that is also amber is 8 (the numerator).
Your turn →
Find .
Tree Highlighted branches
A bag holds 3 red and 2 blue balls. Two are taken out without replacement. Find P(different colours | at least one red).
Your turn →
Find .
Table Two-way table
50 students were asked if they play a school sport. Read off a row or column total for "given".
| Sport | No sport | Total | |
|---|---|---|---|
| Boys | 18 | 7 | 25 |
| Girls | 15 | 10 | 25 |
| Total | 33 | 17 | 50 |
Find P(sport | girl). "Given girl" uses only the Girls row (25), highlighted.
Your turn →
Find .