Different kinds of Mathematical Induction
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1 Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true} i (), we get the Priciple of Mathematical Iductio If () P() is true; () P() is true for some N P() is true the P() is true N (3) Secod Priciple of Mathematical Iductio If () P() is true; () i, P(i) is true [ie P() P() P() is true] P() is true the P() is true N (4) Secod Priciple of Mathematical Iductio (variatio) If () P() P() is true; () P(-) P() is true for some N\{} P() is true the P() is true N (5) Secod Priciple of Mathematical Iductio (variatio) If () P() P() P(m) is true; () P() is true for some N P(m) is true the P() is true N (6) Odd-eve MI If () P() P() is true; () P() is true for some N P() is true the P() is true N Page
2 (7) Bacward MI More difficult types of Mathematical Iductio If () P() is true A, where A is a ifiite subset of N; () P() is true for some N P( ) is true the P() is true N (8) Bacward MI (variatio) (more easily applied tha (7)) If () P() is true; () P( ) is true for some N P( ) is true; (3) P() is true for some N P( ) is true the P() is true N (9) Differet startig poit If () P(a) is true, where a N; () P() is true for some N, where a P() is true the P() is true N\{,,, a } (0) Spiral MI If () P() is true; () P() is true for some N Q() is true Q() is true for some N P() is true the P(), Q() are true N () Double MI Double MI ivolves a propositio P(m, ) with two variables m, If () P(m, ) ad P(, ) is true m, N; () P(m, ) ad P(m, ) are true for some m, N P(m, ) is true the P(m, ) is true m, N Page
3 A Prime Number Theorem [Secod Priciple of Mathematical Iductio] Solutio Prove that the th prime umber p < Let P() be the propositio : p < For P(), p < P() is true Assume P(i) is true i st i, ie p <, p <,, p < (*) For P( ), Multiply all iequalities i (*), pp p < p pp < For ay prime factor p of p pp, we have p < Sice p, p,, p are ot prime factor of p pp, we have p < p ad hece p p p p < P( ) is true By the Secod Priciple of Mathematical Iductio, P() is true Recurrive formula [Secod Priciple of Mathematical Iductio] Let {a } be a sequece of real umbers satisfyig a, a 3 ad a 3a a Prove that a - Solutio Let P() be the propositio : a - For P() P(), a -, a - P() P() is true Assume P() P() is true for some ie a - () a () For P(), a 3a a 3( ) ( - ) P( ) is true By the Secod Priciple of Mathematical Iductio, P() is true Odd Eve Mathematical Iductio Let a, a a a 4 Prove that a ( ) Solutio Page 3
4 4 Let P() be the propositio : a ( ) For P(), a ( ) 4 P() P() is true 4 For P(), a ( ) (*) 4 Assume P() is true for some ie a ( ) For P( ), 4, by (*) a a ( ) ( ) 4 [( ) ] ( ) P( ) is true By the Priciple of Mathematical Iductio, P() is true Bacward Mathematical Iductio Let f() be a cove fuctio defied o [a, b], ie f () f ( ) f for all, [a, b] For each positive iteger, cosider the statemet: I() : If i [a, b], i,,,, the f ( ) f ( ) f (a) Prove by iductio that I( ) is true for every positive iteger (b) Prove that if I() ( ) is true, the I(-) is true (c) Prove that I() is true for every positive iteger Solutio (a) I() : If i [a, b], i,,,, the f () f ( ) f For I( ), sice it is give that f () f ( ) f I( ) is true Assume I( ) is true ie f () f ( ) f () For I( ), f ( ) f ( ) f ( ) f ( ) f f f f f f f f I( ) is true, by (), by I() Page 4
5 (b) Assume I() is true ( ), ie Put f ( ) f ( f ( ) f ( ) f ) f, the f f f () f ( ) f I( ) is also true (c), ( ad r ) such that r Spiral Mathematical Iductio Give a sequece {a } satisfyig a m- 3m(m ) ad a m 3m, where m Let S a i i, prove that S S m m m 4m m 4m ( 3m ) () ( 3m ) () Solutio Let P(m) be the propositio : S m( 4m 3m ) m Q(m) be the propositio : S m( 4m 3m ) m For P(), S a () is true for m Assume P() is true for some, ie S ( 4 3 ) (*) (a) For Q(), S S - a ( 4 3 ) 3 ( 4 3 ) (b) For P( ), S S a ( 4 3 ) 3( ) Q() is true [ ] 3 [( 4 4) ( ( ) ] ) 3 [ 4( ) 3( ) ( ) ] ( ) 4( ) 3( ) [ ] P( ) is true Sice () P() is true () P() is true Q() is true P( ) is true By the Priciple of Mathematical Iductio, P() is true Sice () P() is true Q() is true () Q() is true P( ) is true Q( ) is true By the Priciple of Mathematical Iductio, Q() is true Page 5
6 Mathematical Iductio with parameter, whe a Let f(a, ) 0, whe a >, a N ( ) f a,, whe a ad f(a, ) f ( a, ) f ( a, ), whe a >, a N Prove that f ( a,) Solutio ( )( a ) Let P() be the propositio : f ( a,) () For P(), there are two cases: a ( )( a ) () a Whe a, LHS f(, ) RHS Whe a >, LHS f(a, ) 0 RHS ( a ) a 0 P() is true () Assume P() is true for some, ie f ( a,) ( )( a ) () a For P( ), there are also two cases: Whe a, LHS f(a, ) f(a, ) RHS Whe a >, LHS f(a,) f(a, ) ( )( a ) ( )( a ), by (), f(a,) ad f(a, ) hold a ( a ) ( )( a ) [( a ) a] a ( ) ( )( a ) RHS a P( ) is true By the Priciple of Mathematical Iductio, P() is true Commet If the propositio with atural umber cotais a parameter a, the we eed to apply mathematical iductio for all values of a Page 6
7 Double Mathematical Iductio Solutio Prove that the umber of o-egative itegral solutio sets of the equatio m, m, is f(m, ) ( m ) ( m ) Let P(m, ) be the give propositio () (a) For P(, ), The oly o-egative itegral solutio set of the equatio is oly itself I (), f(, ) ( ) ( ) P(, ) is true For P(m, ), The o-egative itegral solutio sets of the equatio m are (, 0, 0, 0), (0,, 0, ),, (0, 0, 0,, ) There are m sets of solutio altogether I (), f(m, ) ( m ) ( m ) P(m, ) is true m (b) Assume P(m, ) ad P(m, ) are true for some m, ie the umber of o-egative itegral solutio sets of the equatios : m () m m (3) are f(m, ) ( m) ( ) ( m ) ad f(m, ) ( m ) m For P(m, ), The o-egative itegral solutio sets of the equatio : m m (4) may be divided ito two parts : m 0 or m > 0 (i) respectively For m 0, equatio (4) becomes equatio (), ad the umber of o-egative itegral solutio sets is f(m, ) ( m) ( ) ( m ) (ii) For m > 0, replace m by m ad equatio (4) becomes: m m, ad the umber of o-egative itegral solutio sets is f(m, ) ( m ) m The total umber of o-egative itegral solutio sets is ( m) ( ) ( m ) ( m ) m P(m, ) is also true ( m ) ( ) m [( ) m] [( ) ( m ) ] ( ) ( m ) By the Priciple of Mathematical Iductio, P(m, ) is true m, Page 7
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