Book of Proof - Amazon S3

Venn Diagrams 21
1.7 Venn Diagrams
In thinking about sets, it is sometimes helpful to draw informal, schematic
diagrams of them. In doing this we often represent a set with a circle
(or oval), which we regard as enclosing all the elements of the set. Such
diagrams can illustrate how sets combine using various operations. For
example, Figures 1.7(a–c) show two sets A and B which overlap in a
middle region. The sets A ∪ B, A ∩ B and A − B are shaded. Such graphical
representations of sets are called Venn diagrams, after their inventor,
British logician John Venn, 1834–1923.
A ∪ B A ∩ B A − B
A B A B A
B
(a) (b) (c)
Figure 1.7. Venn diagrams for two sets
Though you are not likely to draw Venn diagrams as a part of a proof
of any theorem, you will probably find them to be useful “scratch work”
devices that help you to understand how sets combine, and to develop
strategies for proving certain theorems or solving certain problems. The
remainder of this section uses Venn diagrams to explore how three sets
can be combined using ∪ and ∩.
Let’s begin with the set A ∪ B ∪ C. Our definitions suggest this should
consist of all elements which are in one or more of the sets A, B and C.
Figure 1.8(a) shows a Venn diagram for this. Similarly, we think of A∩B∩C
as all elements common to each of A, B and C, so in Figure 1.8(b) the
region belonging to all three sets is shaded.
C
C
A
A ∪ B ∪ C
B
A
A ∩ B ∩ C
B
(a)
(b)
Figure 1.8. Venn diagrams for three sets