Mathematical Methods for Physics and Engineering - Matematica.NET

1.7 SOME PARTICULAR METHODS OF PROOFA IFF B A is true if and only if B is true or B ⇐⇒ A,A and B necessarily imply each other B ⇐⇒ A.Although at this stage in the book we are able to employ for illustrative purposesonly simple and fairly obvious results, the following example is given as a modelof how necessary and sufficient conditions should be proved. The essential pointis 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, thelines of the second part of the proof will not be simply those of the first writtenin reverse order.◮Prove that (A) a function f(x) is a quadratic polynomial with zeros at x =2and x =3if and only if (B) the function f(x) has the form λ(x 2 − 5x +6) with λ a non-zero constant.(1) Assume A, i.e.thatf(x) is a quadratic polynomial with zeros at x =2andx =3.Letits form be ax 2 + bx + c with a ≠ 0. Then we have4a +2b + c =0,9a +3b + c =0,and subtraction shows that 5a + b =0andb = −5a. Substitution of this into the first ofthe above equations gives c = −4a − 2b = −4a +10a =6a. Thus, it follows thatf(x) =a(x 2 − 5x +6) with a ≠0,and establishes the ‘A only if B’ part of the stated result.(2) Now assume that f(x) has the form λ(x 2 − 5x +6)withλ a non-zero constant. Firstlywe note that f(x) is a quadratic polynomial, and so it only remains to prove that itszeros occur at x =2andx =3.Considerf(x) = 0, which, after dividing through by thenon-zero constant λ, givesx 2 − 5x +6=0.We proceed by using a technique known as completing the square, for the purposes ofillustration, although the factorisation of the above equation should be clear to the reader.Thus we writex 2 − 5x +( 5 2 )2 − ( 5 2 )2 +6=0,(x − 5 2 )2 = 1 , 4x − 5 = ± 1 .2 2The two roots of f(x) = 0 are therefore x =2andx =3;thesex-values give the zerosof f(x). This establishes the second (‘A if B’) part of the result. Thus we have shownthat the assumption of either condition implies the validity of the other and the proof iscomplete. ◭It should be noted that the propositions have to be carefully and preciselyformulated. If, for example, the word ‘quadratic’ were omitted from A, statementB would still be a sufficient condition for A but not a necessary one, since f(x)could then be x 3 − 4x 2 + x + 6 and A would not require B. Omitting the constantλ from the stated form of f(x) inB has the same effect. Conversely, if A were tostate that f(x) =3(x − 2)(x − 3) then B would be a necessary condition for A butnot a sufficient one.35