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

62 The Nuts and Bolts of Proof, Third Edition
EXAMPLE 1. The number, usually indicated by 1, such that:
axl=lxa=a
for all real numbers a is unique. (The number 1 is called the identity for
multiplication of real numbers.)
Proof: Let t be a number with the property that:
ax t = t X a = a
for all real numbers a (even for a=l and for a = t). (We cannot use the
symbol 1 for this number, because as far as we know t could be different
from 1.)
Because 1 leaves all other numbers unchanged when multiplied by them,
we have
1 xt = t.
Because t leaves all other numbers unchanged when multipHed by them, we
have:
Therefore,
1 xf = 1.
t=lxt=l.
This proves that t=l. Thus, it is true that only the number 1 has the
required properties (i.e., the identity element for multipHcation is
unique). •
Very often existence and uniqueness theorems are combined in statements
of the form: "There exists a unique ..." The proof of this kind of statements
has two parts:
1. Prove the existence of the object described in the statement.
2. Prove the uniqueness of the object described in the statement.
While these two steps can be performed in any order, it seems to make sense
to prove the existence of an object before proving its uniqueness. After all, if
the object does not exist, its uniqueness becomes irrelevant. If the object can
be constructed explicitly (to prove its existence), the steps used in the
construction might provide a proof of its uniqueness.