Operations on Sets
How to Do set operations with a Venn diagram
Use a Venn diagram to work out intersections, unions and combined set operations, especially in worded problems.
Before you start
- Know the symbols ∪ (union), ∩ (intersection) and the complement A′.
- Understand that the universal set is the total of all elements.
- Be able to add and subtract whole numbers and solve a simple linear equation.
When to use it
Use a Venn diagram to work out intersections, unions and combined set operations, especially in worded problems.
The steps
- Draw the universal set as a rectangle and each set as an overlapping circle inside it.
- Fill in what you know, starting with the overlap (the intersection).
- Work out the remaining regions from the totals given.
- Shade the region the question asks for.
- Count or list the elements in the shaded region.
Worked example
In a class of 40 students, 22 take Additional Mathematics, 18 take Physics and 10 take both. How many take Additional Mathematics or Physics, and how many take neither?
- Draw the universal set (40 students) as a rectangle, with two overlapping circles A for Additional Mathematics and P for Physics.
- Fill in the overlap first: the number taking both is 10, so write 10 in the intersection.
- Work out the rest from the totals: only A = 22 − 10 = 12, only P = 18 − 10 = 8, and neither = 40 − (12 + 10 + 8) = 40 − 30 = 10.
- Shade the region for 'Additional Mathematics or Physics' both circles together (the union).
- Count the shaded region: 12 + 10 + 8 = 30 students take at least one subject, and 10 students are left outside.
A second example, with a twist
This time the number taking both is unknown, so you form and solve a linear equation instead of subtracting directly. In a group of 50 people, 30 like tea, 25 like coffee and 5 like neither.
If x people like both, find the value of x.
- Draw the universal set (50 people) as a rectangle with two overlapping circles T for tea and C for coffee.
- Fill in the overlap first with the unknown: write x in the intersection (people who like both).
- Work out the rest in terms of x: only tea = 30 − x, only coffee = 25 − x, and neither = 5; the four regions add to 50, so (30 − x) + x + (25 − x) + 5 = 50, giving 60 − x = 50, so x = 10.
- Shade the region the question asks for, the intersection (people who like both).
- Count the shaded region: x = 10, so 10 people like both.
Practise this in a KBAT problem
Frequently asked questions
Why do I fill in the overlap first?
The overlap (the intersection) belongs to both circles, so it is already counted inside each subject total. If you fill the outer parts first you will double-count these people.
Writing the 'both' number in the middle first, then subtracting it from each total, gives the correct 'only A' and 'only B' figures.
Where do I put people who are in neither set?
Write that number inside the rectangle but outside both circles. It still belongs to the universal set, so it counts towards the grand total.
To find it, subtract everyone inside the circles from the universal total: neither = n(ξ) − n(A ∪ B). Forgetting this region is a common way to lose marks.
When do I use the formula instead of drawing?
The formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) gives the same answer as the diagram and is quick when you only need the union. Draw the Venn diagram when the question has several regions, a 'neither' group, or an unknown you must solve for, it is much easier to see.