5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR + 155
be possible, if not practical, to prove the theorem by enumerating all the
cases. We proved theorems of the propositional calculus in this manner in
Chapter 2. A truth table merely enumerates all the possible truth combina-
tions (2" corresponding to n letters) for the simple statements that comprise
a given compound statement form. Recall, in this connection, Exercises 9
and 1 l(c), Article 1.5. We deal with another example of this type in Exer-
cise 16.
As we conclude this article and you begin the exercises, there are two
general principles to keep in mind: (1) It is essential in doing any proof,
especially proofs by transitivity, to have a clear picture of what has come
before-that is, what axioms, previously proved theorems, or any facts that
are taken to be "known," are available for use in the proof. (2) It is important
to approach theorem-proving in an active way: Always have pen and paper
in front of you. Don't waste time staring at the book; write things down in-
stead. In particular, write down the desired conclusion and the hypothesis
(if any) and write a list of any definitions and known relationships that may
be relevant. Don't be discouraged if your first approach doesn't work; be
flexible and willing to try a number of approaches.
If the logical structure of the conclusion to be derived is more complex
than we have considered thus far, you will still be faced with the question,
"How do I go about starting this proof?" Read on, for forthcoming articles
will deal directly with this question, in a variety of situations.
Exercises
- Throughout this exercise, make use of the associative and commutative laws for
addition and multiplication of real numbers, as well as the law of distributivity of
multiplication over addition. Write explicitly the justification for each step. Prove
that:
(a) (a + b)2 = a2 + 2ab + b2 Va, b E R
(6) (a + b)(a - b) = a2 - b2 Va, b E R
(c) [a + (b + c)] + d = a + [b + (c + d)] Va, b, c, d E R
(d) a(bc) = c(ba) Va, b, c E R
(e) (ab+ad)+(cb+cd)=(a+c)(b+d) Va,b,c,d~R
(f) a(b + c + d) = ab + ac + ad Va, b, c, d E R - (a) Use elementary trigonometric identities (e.g., double angle formulas, defini-
tion of tan x in terms of sin x and cos x, etc.) to verify these trigonometric identities:
cos4 x - sin4 x = cos 2x Vx E R
4 sin3 x cos x = sin 2x - sin 2x cos x Vx E R
sec x - sin x tan x = cos x Vx E R such that cos x # 0
(tan x - l)/(tan x + 1) = (1 - cot x)/(l + cot x) Vx E R such that
sin x # 0 and cos x # 0.
Write out a complete proof by transitivity for the identity of Example 4.