MathematicsExam code: 0580

Set Notation and Venn Diagrams

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How the diagram is built

A Venn diagram shows sets as circles inside a rectangle, where the rectangle is the universal set
Overlapping circles share the elements that lie in both sets
Anything belonging to no named set is written inside the rectangle but outside every circle
This specification limits Venn diagrams to two or three sets

Venn diagram with universal set as a rectangle, labelled here with the alternative symbol xi, containing two overlapping circles labelled A and B; 12, 6 and 2 lie in A alone, 14 and 28 in the overlap, 7, 21 and 35 in B alone, and 8, 5 and 1 outside both circles — Source: Set notation by Save My Exams

Regions and operations

Each operation corresponds to a region of the diagram, which is what makes the notation easy to apply
The union covers both circles entirely; the intersection covers only the overlap

Three Venn diagrams, each a rectangle labelled ℰ (script E) holding two overlapping circles A and B. In the first both circles are shaded, labelled A union B, A or B or both. In the second only the overlap is shaded, labelled A intersection B, A and B. In the third everything inside the rectangle except circle A is shaded, labelled A complement, not A — Source: Set notation by Save My Exams

A complement shades everything outside the named set but still inside the rectangle
Combined expressions are read from the inside out, so (A ∪ B)′ means shade everything the union does not cover

Counting from a diagram

n(A) counts every element inside circle A, including those in the overlap
Elements in the overlap belong to both sets and are counted once in a union

Worked example

Placing elements in a three-set Venn diagram

ℰ = {x : x is an integer, 1 ⩽ x ⩽ 15}, A = {prime numbers}, B = {odd numbers} and C = {multiples of 3}. Complete the Venn diagram for A, B and C, then find n(A ∩ C ∩ B′).

Solution:

List each set from ℰ first: A = {2, 3, 5, 7, 11, 13}, B = {1, 3, 5, 7, 9, 11, 13, 15} and C = {3, 6, 9, 12, 15}
Fill the centre first: 3 is the only number in all three sets
In A and B but not C: 5, 7, 11, 13
In B and C but not A: 9, 15
In A and C but not B: nothing, because 3 is the only prime that is a multiple of 3 and it is already in the centre
A only: 2, the only even prime; B only: 1, which is odd but not prime; C only: 6, 12
Outside all three circles: 4, 8, 10, 14
Check the count: 1 + 4 + 2 + 0 + 1 + 1 + 2 + 4 = 15, so every element of ℰ is placed exactly once
n(A ∩ C ∩ B′) counts the empty region, so n(A ∩ C ∩ B′) = 0, a number, not ∅

The completed Venn diagram: 3 in the centre where all three circles overlap; 5, 7, 11 and 13 where A and B overlap outside C; 9 and 15 where B and C overlap outside A; nothing where A and C overlap outside B; 2 in A only; 1 in B only; 6 and 12 in C only; and 4, 8, 10 and 14 inside the rectangle outside all three circles

Worked example

Counting from a Venn diagram

In a group of 30 students, 18 study art (A) and 14 study biology (B). 7 study both. Show this on a Venn diagram, then work out how many study neither.

Solution:

Fill the overlap first, because the two given totals each already include it
The intersection holds 7
Art only = 18 − 7 = 11
Biology only = 14 − 7 = 7
Those three regions hold 11 + 7 + 7 = 25 students
The universal set holds 30, so the region outside both circles holds 30 − 25 = 5
Check by adding all four regions: 11 + 7 + 7 + 5 = 30
Writing 18 in the art circle and 14 in the biology circle would count the 7 twice and total 39

The completed Venn diagram: a rectangle labelled ℰ holding overlapping circles A and B, with 11 in A only, 7 in the overlap, 7 in B only and 5 inside the rectangle outside both circles

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