Page 132 - HOW TO PROVE IT: A Structured Approach, Second Edition

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118 Proofs
Proofs involving the quantifiers for all and there exists are often difficult for
them.
Thatlastsentenceconfusedyou,didn’tit?You’reprobablywondering,“Who
are they?” Readers of your proofs will experience the same sort of confusion
if you use variables without explaining what they stand for. Beginning proof-
writers are sometimes careless about this, and that’s why proofs involving the
quantifiers for all and there exists are often difficult for them. (It made more
sense that time, didn’t it?) When you use the strategies we’ve discussed in this
section, you’ll be introducing new variables into your proof, and when you
do this, you must always be careful to make it clear to the reader what they
stand for.
For example, if you were proving a goal of the form ∀x ∈ AP(x), you would
probably start by introducing a variable x to stand for an arbitrary element of
A. Your reader won’t know what x means, though, unless you begin your proof
with “Let x be an arbitrary element of A,” or “Suppose x ∈ A.” Of course,
you must be clear in your own mind about what x stands for. In particular,
because x is to be arbitrary, you must be careful not to assume anything about
x other than the fact that x ∈ A. It might help to think of x as being chosen
by someone else; you have no control over which element of A they’ll pick.
Using a given of the form ∃xP(x) is similar. This given tells you that you
can introduce a new variable x 0 into the proof to stand for some object for
which P(x 0 ) is true, but you cannot assume anything else about x 0 .Onthe
other hand, if you are proving ∃xP(x), your proof will probably start “Let
x = ...” This time you get to choose the value of x, and you must tell the
reader explicitly that you are choosing the value of x and what value you have
chosen.
It’s also important, when you’re introducing a new variable x, to be sure you
know what kind of object x is. Is it a number? a set? a function? a matrix? You’d
better not write a ∈ X unless X is a set, for example. If you aren’t careful about
this,youmightendupwritingnonsense.Youalsosometimesneedtoknowwhat
kind of object a variable stands for to figure out the logical form of a statement
involving that variable. For example, A = B means ∀x(x ∈ A ↔ x ∈ B)if A
and B are sets, but not if they’re numbers.
The most important thing to keep in mind about introducing variables into
a proof is simply the fact that variables must always be introduced before they
are used. If you make a statement about x (i.e., a statement in which x occurs
as a free variable) without first explaining what x stands for, a reader of your
proof won’t know what you’re talking about – and there’s a good chance that
you won’t know what you’re talking about either!