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18 The Nuts and Bolts of Proof, Third Edition
Note: The number of digits used in Example 5 is irrelevant. This is a special
case of the much more general statement: "A number is divisible by 3 when
the sum of its digits is divisible by 3." We chose to use five digits because the
proof of the more general statement, which at the beginning is very similar to
the one above, can be easily completed using a technique that will be
introduced later—namely, "proof by induction." Let's look at the setup of
the general proof.
Let n be an integer number with n = atak-i... fl2<^i^o, 0 < a/ < 9 for all
i = 0, 1, 2,..., k and ak ^ 0, such that ak + ak-i H h ^2 + ai + ao == 3r,
where t is an integer number. Then, following the same steps performed in
the proof in Example 6, we can write:
n = lO^Uk + 10^"^a/c-i H
h 10^^2 + lOai + ao
= lO^ak + 10^~^ak-i H h 10^^2 + lOai + (3t-ak-ak-\ ^2 -«i)
= (10^ - IK + (10^-^ - l)ak-x + • • • + 99^2 + 9ai + 3t.
At this point, to be able to show that n is divisible by 3, we need to prove
that 10^ — 1 is divisible by 3 for all s>\. This is the step that can require
proof by induction (see Exercise 8 at the end of the section on Mathematical
Induction), unless one is famihar with modular arithmetic. As already
mentioned, as one's mathematical background increases, one has more tools
to use and therefore becomes able to construct the proof of a statement using
several different approaches.
EXAMPLE 6. Let/and g be two real-valued functions defined for all real
numbers and such that f o g is well defined for all real numbers. If both
functions are one-to-one, then / o ^ is a one-to-one function.
Discussion: We will separate the hypothesis and the conclusion:
A. We are considering two functions that have the following properties:
1. They are defined for all real numbers.
2. They are one-to-one.
The fact that the functions are called / and g is irrelevant. We can use
any two symbols, but having a quick way to refer to the functions
does simplify matters.
B: The function is one-to-one.
We can fully understand the meaning of the given statement only if we are
famihar with the definitions of function, one-to-one function, and composition
of functions. (See the section on facts and properties of functions at the
front of the book.)