Operations on Sets
How to Describe a shaded set region
Use this to write the set notation for a shaded region of a Venn diagram.
Before you start
- Know the notation A ∩ B (intersection), A ∪ B (union) and the complement A′.
- Be able to picture the four basic regions of a two-set Venn diagram.
- Understand the universal set ξ and what a complement means.
When to use it
Use this to write the set notation for a shaded region of a Venn diagram.
The steps
- Identify which basic regions are shaded (only A, only B, the overlap, outside both).
- Match each region to a set operation: intersection, union or complement.
- Combine the operations to describe the whole shaded area.
- Check by shading your notation back onto a blank diagram.
- Simplify the notation if a shorter equivalent exists.
Worked example
In a Venn diagram with universal set ξ and sets A and B, both circles are fully shaded, including the overlap, but the region outside the circles is not shaded. Write the set notation for the shaded region.
- Identify which basic regions are shaded: the 'only A' part, the overlap, and the 'only B' part are all shaded; the region outside both is not.
- Match each region to an operation: 'only A' plus the overlap plus 'only B' together make up everything inside at least one circle.
- Combine the operations: everything inside A or B (or both) is the union, A ∪ B.
- Check by shading A ∪ B onto a blank diagram: it covers both whole circles and nothing outside, this matches.
- Simplify: A ∪ B is already the shortest form, so no change is needed.
A second example, with a twist
This time the shaded area is in two separate pieces, so you must combine them and then simplify to a single, shorter notation. In a Venn diagram with sets A and B, the 'only A' region and the region outside both circles are shaded, while the overlap and the 'only B' region are left blank.
Write the set notation for the shaded region in its simplest form.
- Identify which basic regions are shaded: the 'only A' region and the region outside both circles; the overlap and 'only B' are blank.
- Match each region to an operation: 'only A' is A ∩ B′ (in A but not B), and outside both circles is (A ∪ B)′.
- Combine the operations: the shaded area is (A ∩ B′) ∪ (A ∪ B)′.
- Check by shading onto a blank diagram: every part except circle B is shaded, that is exactly everything not in B.
- Simplify: everything that is not inside B is the complement of B, so the notation reduces to B′.
Practise this in a KBAT problem
Frequently asked questions
How do I write 'the region outside both circles'?
That region is everything in the universal set that is not in A and not in B, so you write it as (A ∪ B)′, the complement of the union. It can also be written A′ ∩ B′, which shades exactly the same area.
Either form is acceptable as long as it matches the diagram.
Are A ∩ B′ and B ∩ A′ the same thing?
No. A ∩ B′ is the part of circle A that lies outside B, the 'only A' region.
B ∩ A′ is the part of circle B that lies outside A, the 'only B' region. They are different pieces of the diagram, so check carefully which circle the shading sits inside before you write it.
How can I be sure my notation is correct?
Shade your answer onto a fresh blank diagram, region by region, and compare it with the original. If every shaded and unshaded part matches, your notation is right.
If it does not, adjust the operations and try again. This check catches most mistakes and takes only a few seconds.