Venn diagrams and set notation
Venn diagrams
A sorts things into overlapping circles inside a rectangle. The overlap holds the things in both circles; the space outside the circles holds those in neither.
- Fill in the overlap first.
- Subtract it to find each "only" region.
- Subtract everything from the total to find "neither".
Example: of 30 students, 18 play football, 12 play tennis and 5 play both. Find P(a student plays exactly one of the sports).
- Both: 5
- Football only: 18 − 5 = 13. Tennis only: 12 − 5 = 7
- Neither: 30 − 13 − 5 − 7 = 5
- Exactly one: 13 + 7 = 20, so P = 20/30 = 2/3
Key terms
Set notation
| Symbol | Meaning |
|---|---|
| ξ | the : everything being considered |
| ∈ | "is a member of": 4 ∈ A means 4 is in set A |
| A ∪ B | the : in A or B or both |
| A ∩ B | the : in both A and B |
| A′ | the : not in A |
Example: ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = even numbers and B = multiples of 3. List A ∩ B, A ∪ B and A′.
- A = {2, 4, 6, 8, 10} and B = {3, 6, 9}
- A ∩ B = {6}
- A ∪ B = {2, 3, 4, 6, 8, 9, 10}
- A′ = {1, 3, 5, 7, 9}
Key terms
Harder Venn diagrams (Higher tier only)
When the overlap is unknown, call it x and form an equation. With three sets, fill in the middle (all three) first, then each pair, then each "only".
Example: 40 people were asked about drinks. 25 like tea, 22 like coffee and 6 like neither. How many like both?
- Let x like both. Then tea only = 25 − x and coffee only = 22 − x
- Everyone adds up to 40: (25 − x) + x + (22 − x) + 6 = 40
- 53 − x = 40
- x = 13
Check: tea only 12, both 13, coffee only 9, neither 6, and 12 + 13 + 9 + 6 = 40.
Conditional probability from a Venn diagram (Higher tier only)
For P(A given B), you already know the item is in B, so only B's circle counts.
P(A given B) = number in both ÷ number in B
Example: using the football and tennis Venn diagram (13 football only, 5 both, 7 tennis only, 5 neither), a student who plays tennis is chosen. Find P(they also play football).
- Tennis players: 5 + 7 = 12
- Of these, 5 also play football
- P(football given tennis) = 5/12
Common mistake
Divide by the number in the circle you're given (12), not the total (30).
Set notation for inequalities (Higher tier only)
Set notation can describe a range of numbers: {x : x > 3} means "the set of all x such that x is greater than 3".
- Read the colon as "such that".
- List the members if the set is small, or keep the inequality if it's infinite.
Example: list the members of {x : x is an integer, −2 ≤ x < 3}.
- Integers from −2 (included) up to 3 (not included)
- {−2, −1, 0, 1, 2}
Example: write "x is less than 1 or greater than 5" in set notation.
- {x : x < 1} ∪ {x : x > 5}