Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.7 SOME PARTICULAR METHODS OF PROOF

AIFFBAis true if and only ifBis trueor B⇐⇒ A,

AandBnecessarily imply each other B⇐⇒ A.

Although at this stage in the book we are able to employ for illustrative purposes

only simple and fairly obvious results, the following example is given as a model

of how necessary and sufficient conditions should be proved. The essential point

is that for the second part of the proof (whether it be the ‘necessary’ part or the

‘sufficient’ part) one needs to start again from scratch; more often than not, the

lines of the second part of the proof willnotbe simply those of the first written

in reverse order.

Prove that (A) a functionf(x)is a quadratic polynomial with zeros atx=2andx=3

if and only if (B) the functionf(x)has the formλ(x^2 − 5 x+6)withλa non-zero constant.

(1) AssumeA,i.e.thatf(x)isa quadratic polynomial with zeros atx=2andx=3.Let
its form beax^2 +bx+cwitha= 0. Then we have

4 a+2b+c=0,

9 a+3b+c=0,

and subtraction shows that 5a+b=0andb=− 5 a. Substitution of this into the first of
the above equations givesc=− 4 a− 2 b=− 4 a+10a=6a. Thus, it follows that

f(x)=a(x^2 − 5 x+6) with a=0,

and establishes the ‘Aonly ifB’ part of the stated result.

(2) Now assume thatf(x)hasthe formλ(x^2 − 5 x+6) withλa non-zero constant. Firstly
we note thatf(x) is a quadratic polynomial, and so it only remains to prove that its
zeros occur atx=2andx=3.Considerf(x) = 0, which, after dividing through by the
non-zero constantλ,gives

x^2 − 5 x+6=0.
We proceed by using a technique known ascompleting the square, for the purposes of
illustration, although the factorisation of the above equation should be clear to the reader.
Thus we write

x^2 − 5 x+(^52 )^2 −(^52 )^2 +6=0,

(x−^52 )^2 =^14 ,

x−^52 =±^12.

The two roots off(x) = 0 are thereforex=2andx=3;thesex-values give the zeros
off(x). This establishes the second (‘AifB’) part of the result. Thus we have shown
that the assumption of either condition implies the validity of the other and the proof is
complete.

It should be noted that the propositions have to be carefully and precisely

formulated. If, for example, the word ‘quadratic’ were omitted fromA, statement

Bwould still be a sufficient condition forAbut not a necessary one, sincef(x)

could then bex^3 − 4 x^2 +x+ 6 andAwould not requireB. Omitting the constant

λfrom the stated form off(x)inBhas the same effect. Conversely, ifAwere to

state thatf(x)=3(x−2)(x−3) thenBwould be a necessary condition forAbut

not a sufficient one.