Page 122 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/
0521861241c03 CB996/Velleman October 20, 2005 2:42 0 521 86124 1 Char Count= 0

108 Proofs
3.3. Proofs Involving Quantifiers

Look again at Example 3.2.3. In that example we said that x could be any-
thing at all, and we proved the statement x ∈ A \ C → x ∈ B. Because the
reasoning we used would apply no matter what x was, our proof actually shows
that x ∈ A \ C → x ∈ B is true for all x. In other words, we can conclude
∀x(x ∈ A \ C → x ∈ B).
This illustrates the easiest and most straightforward way of proving a goal
of the form ∀xP(x). If you can give a proof of the goal P(x) that would work
no matter what x was, then you can conclude that ∀xP(x) must be true. To
make sure that your proof would work for any value of x, it is important to
start your proof with no assumptions about x. Mathematicians express this by
saying that x must be arbitrary. In particular, you must not assume that x is
equal to any other object already under discussion in the proof. Thus, if the
letter x is already being used in the proof to stand for some particular object,
then you cannot use it to stand for an arbitrary object. In this case you must
choose a different variable that is not already being used in the proof, say y,
and replace the goal ∀xP(x) with the equivalent statement ∀yP(y). Now you
can proceed by letting y stand for an arbitrary object and proving P(y).

To prove a goal of the form ∀xP(x):
Let x stand for an arbitrary object and prove P(x). The letter x must be a
new variable in the proof. If x is already being used in the proof to stand for
something, then you must choose an unused variable, say y, to stand for the
arbitrary object, and prove P(y).

Scratch work
Before using strategy:

Givens Goal
— ∀xP(x)
—
After using strategy:
Givens Goal
— P(x)
—
Form of final proof:

Let x be arbitrary.
[Proof of P(x) goes here.]
Since x was arbitrary, we can conclude that ∀xP(x).