Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.3 PROOF BY SPECIALIZATION AND DIVISION INTO CASES 171

to write out the proofs. Don't, however, become discouraged if you can't
do them at this stage. If you can prove the symmetry property in Example
1, then you already have an implicit grasp of a proof technique known as
specialization. If you can handle Example 2 now, then you are able to
deal with a proof that calls for a division into cases.
These two methods will be useful throughout the remainder of the text.
Regardless of the logical form of the desired conclusion of a theorem, these
two methods are basic tools of the mathematician for adapting given
hypotheses toward that conclusion.

SPECIALIZATION

Repeatedly we have stressed the fact that a general assertion (Vx)(p(x))

cannot be proved by verifying a particular instance p(a), where a is a specific

element of the domain of discourse of p(x). But frequently, in deriving a

conclusion on the basis of an assumption or hypothesis (Vx)(p(x)), we find

that a particular case of the latter proves to be just what is needed to get the

desired result. In such situations the special case may involve either the

substitution of a specific constant a for the variable x (see Example 3) or

the replacement of x by some expression involving an arbitrary quantity

y whose value was fixed as a part of the initial setting up of the proof (see

Example 4).

The first exposure most students get to proofs involving the method of

specialization is in proofs of certain theorems in trigonometry.

EXAMPLE 3 Suppose it is known (i.e., has been assumed as an axiom or

has already been proved) that sin (x + a) = sin x cos a + cos x sin a for

all real numbers x and a. Prove that sin (x + (42)) = cos x, for all real

numbers x.

Solution Let x be an arbitrary real number. Consider the special case

a = n/2 of the known identity. This gives

sin (x + (42)) = sin x cos (n/2) + cos x sin (42)

= (sin x)(O) + (cos x)(l)

= cos x, as desired.

EXAMPLE 4 Suppose it is known that sin x = cos ((7~12) - x) for all real

numbers x. Use this result to prove that cos x = sin ((~12) - x) for all

real numbers x.

Solution Let x be an arbitrary real number. Recalling that sin x' =

cos ((42) - x') for any real number x' is known to be true, consider the

quantity sin ((7~12) - x). Letting x' = (42) - x in the equation of the

previous sentence, we have sin ((7112) - X) = sin x' = cos ((42) - x') =

cos ((42) - ((42) - x)) = cos x, which is precisely what we wanted to

prove. 0