Sets, Set Notation and Venn Diagrams
A set is a collection of distinct elements, written inside curly brackets. The universal set, written with the symbol for xi, contains every element under consideration in that question, and it is drawn as the rectangle enclosing a Venn diagram. A' is the complement of A: everything in the universal set that is not in A. The intersection of A and B is the set of elements in both, and the union of A and B is the set of elements in at least one of them. The empty set contains no elements. n(A) means the number of elements in A. Set-builder notation such as the set of x where x is an integer and 2 is less than x which is less than 9 describes a set by a rule rather than by listing. Level 2 Further Maths expects fluent reading and writing of these symbols, shading a region of a Venn diagram that matches an expression such as A intersect B complement, and using the diagram to answer probability questions, where the probability of an event is the number of favourable elements over the total in the universal set.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Draw the rectangle for the universal set first, then the circles, before you write a single number. The rectangle is where every stray element ends up.
- Fill a Venn diagram from the middle outwards. Put the number in the intersection first, then subtract it from each individual total to get the numbers in the outer parts of each circle.
- Check the total. The four regions of a two-set diagram must add up to n of the universal set; if they do not, the intersection is wrong.
- Translate a set expression into words before shading. Intersection means and, union means or, and a complement means everything except. Read the expression from the inside of the brackets outwards.
- For a complement of a compound region, shade the compound region lightly first, then shade everything else as your answer. Trying to shade the complement directly is where most errors happen.
- For probability from a Venn diagram, count the elements in the region asked for and divide by the total in the universal set. Read carefully whether the question restricts you to a smaller pool, such as given that the element is in B.
Worked example
In a group of 40 students, 24 study French, 18 study German and 9 study both. Draw the Venn diagram, then find n(F union G), n(F' intersect G') and the probability that a student picked at random studies French but not German.
- Draw a rectangle for the 40 students, with two overlapping circles labelled F and G inside it.
- Start in the middle: 9 students study both, so write 9 in the intersection.
- Work outwards for French: 24 study French in total, and 9 of them are already counted in the middle, so 24 - 9 = 15 go in the French-only region.
- Do the same for German: 18 - 9 = 9 go in the German-only region.
- Find what is left over: 15 + 9 + 9 = 33 students are inside the circles, so 40 - 33 = 7 students study neither. Write 7 in the rectangle outside both circles.
- Read off the answers. n(F union G) is everything in at least one circle: 15 + 9 + 9 = 33. n(F' intersect G') is the region outside both, which is 7.
- French but not German is the French-only region, 15 students, so the probability is 15/40, which simplifies to 3/8.
Practice questions
Try each question, then tap to reveal the answer.
Q1Given the universal set is the integers from 1 to 12, A is the set of even numbers and B is the set of multiples of 3, list A intersect B and A union B.Show answer
Answer: A = {2,4,6,8,10,12}, B = {3,6,9,12}. A intersect B = {6,12}. A union B = {2,3,4,6,8,9,10,12}.
Q2For the same universal set and sets as above, list A' and state n(A').Show answer
Answer: A' = {1,3,5,7,9,11}, so n(A') = 6.
Q3Write in set notation: the set of integers x such that x is greater than -3 and less than or equal to 4.Show answer
Answer: {x : x is an integer, -3 < x <= 4}, which lists as {-2,-1,0,1,2,3,4}.
Q4n(A) = 17, n(B) = 12 and n(A intersect B) = 5. Calculate n(A union B).Show answer
Answer: n(A union B) = 17 + 12 - 5 = 24. The intersection is subtracted because those 5 elements were counted in both totals.
Q5Describe in words the region represented by (A union B)'.Show answer
Answer: Everything that is in neither A nor B: the part of the universal set outside both circles.
Q6Explain why A intersect A' is always the empty set.Show answer
Answer: A' contains exactly the elements that are not in A, so no element can be in both A and A' at once. The intersection therefore has no elements.
Q7In a class of 30, 21 own a bike and 14 own a scooter. Every student owns at least one. How many own both?Show answer
Answer: 21 + 14 = 35, which exceeds 30 by 5, so 5 students were counted twice and therefore own both.
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
The universal set is the integers from 1 to 20. A is the set of factors of 20, B is the set of prime numbers and C is the set of multiples of 4. (a) List the elements of each set. (b) Find n(A intersect B) and list A intersect C. (c) List (B union C)'.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 7 available
In a survey of 120 people, 68 read a newspaper, 54 listen to a news podcast and 22 do both. A person is chosen at random. (a) Draw a Venn diagram to represent this information. (b) Find the probability that the person does neither. (c) Given that the person reads a newspaper, find the probability that they also listen to a podcast.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 7 available
See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the sets, set notation and venn diagrams worksheet pack - 12 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on sets, set notation and venn diagrams, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.