epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 73
We will start by constructing the set Ari(BU
darkest shaded part in Diagram 4:
C), represented by the
Diagram 4
We now consider the representation of the set (yl fi 5) U (^ fi C) shown m
Diagram 5:
Diagram 5
Therefore, the two sets obtained using the Venn diagrams are equal.
The use of Venn diagrams does not provide a proof, but it offers a
good illustration. Venn diagrams have the same role of examples. Moreover
they become difficult to work with when the number of sets depicted
becomes larger—for example, when deahng with arbitrary collections of
four or more sets.
EXAMPLE 4. Let .4 = {x G Z | x is a multiple of 5} and J5 = {x€Z|xisa
multiple of 7}. Then,
Proof:
APiB = {x eJ.\x\sdi multiple of 35}.
Part 1. ^nBc{xGZ|xisa multiple of 35}.
Let X e Ar^B. Then x e A and x eB. This implies that x is a multiple
of 5 and it is a multiple of 7. Therefore, x = 5n and x = 7m, where n and m
are integer numbers.