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

Introduction and Basic Terminology 19
By definition, a real-valued function h is said to be one-to-one if for every
two real numbers xi and xj such that xi 7^X2, it follows that h{x\) ^ /ife).
The composition of two functions is the function defined as
fog{x)=f{g{x)).
Using these two definitions, we can explicitly rewrite the conclusion to be
reached as:
B. If x\ and X2 are two real numbers such that x\ ^ X2, then
fog{xi)^fog{x2).
Proof: Let x\ and X2 be two real numbers such that Xi ^ X2. (Examine in
detail each step of the construction of the function to see "what happens" to
the values corresponding to x\ and X2.) As ^ is a one-to-one function, it
follows that:
Set yi = g(xi) and y2 = g(x2).
g{xi) ^ g(x2y
As/is a one-to-one function and yi 7^ y2, it follows that:
f(yi)
^fiyi)
thatis,/(^(xi))7^/(^(x2)).
Thus, by the definition of, we can conclude that if x\ and X2 are two real
numbers such that x\ ^ X2, then
Therefore is / o ^ a one-to-one function.
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RELATED STATEMENTS
If we are having a conversation with a person who makes a statement
whose meaning escapes us, we might ask, "What do you mean?" Our hope is
that the wording of the statement will be changed so that we can grasp its
meaning, without changing the meaning itself. This situation happens when
we are working with mathematical statements as well. How do we change a
mathematical statement into one that is easier for us to handle but has the
same mathematical meaning?
Two statements are logically equivalent when they have the same truth
table. So, we can change a mathematical statement into one easier to work