Principle of Mathematical Induction

Transcription

1 Secto. Prcple of Mthemtcl Iducto.. Defto Mthemtcl ducto s techque of proof used to check ssertos or clms bout processes tht occur repettvely ccordg to set ptter. It s oe of the stdrd techques of proof computer scece. Cosder the followg exmple: Exmple: Prove > for ll turl umber. (Note, usg predcte logc otto, ths could be wrtte: Prove, > ) Let Clm() be >. For, Clm() s > LHS, RHS 0 LHS > RHS, thus Clm() s true. For, Clm() s > LHS 8, RHS LHS > RHS, thus Clm() s true. WUCT Numbers 44

2 For, Clm() s > LHS 7, RHS 4 LHS > RHS, thus Clm() s true. Ad so o for 4, 5, 6, How do we prove Clm() for ll other vlues of? The process bove wll ever prove Clm() for ALL turl umbers. The Prcple of Mthemtcl Iducto (PMI) provdes method for provg such clms d results from the wellorderg property o. The PMI provdes method of proof to use whe the sttemet to be proved s of the form:, P( ). WUCT Numbers 45

3 .. Bsc Prcple Theorem: The Prcple of Mthemtcl Iducto (PMI) For ll turl umbers, let Clm() be sttemet. If. Clm() s true, d. For ll turl umbers k, f Clm(k) s true, the Proof : Clm( k ) s lso true, the Clm() s true for ll turl umbers, tht s,. Let T { t : Clm() t s flse}. If we c show tht T s empty, the Clm() s true for ll. Note: T s subset of Õ. Also, we kow the followg: Clm() s true. For ll turl umbers k, f Clm(k) s true, the Clm( k ) s lso true. Assume T s ot empty. The T hs lest elemet. [Why?] Let ths lest elemet be t 0, so Clm( t 0 ) s flse. (*) WUCT Numbers 46

4 By sttemet of the PMI, t 0, so t 0 s turl umber. Also, t s ot T. 0 Therefore, Clm( t 0 ) s true. However, by sttemet of the PMI, Clm( t 0 ) true mples Clm( t 0 ) Clm( t 0 ) s true. Ths s cotrdcto wth sttemet (*). So, our ssumpto tht T s ot empty must be flse. Hece, T must be empty. Therefore, Clm() s true for ll. Exercse: Rewrte the Prcple of Mthemtcl Iducto, usg predcte logc. WUCT Numbers 47

5 Exmple: ( ) Prove K for ll. ( ) Let Clm() be K. Does the clm stsfy the codtos of the Prcple of Mthemtcl Iducto?. Is Clm() true? ( ) Clm() s LHS ; RHS Therefore, LHS RHS d so Clm() s true. WUCT Numbers 48

6 . We must show tht for k, Clm(k) true Clm( k ) true. Assume Clm(k) s true, tht s, ( k ) k K k K(). Prove Clm( k ) s true, tht s, K k LHS ( k ) k ( k ) K k k k k ( k ) k k ( k )( k ) RHS Therefore, LHS RHS. ( k )( k ) ( k ) by () Therefore, Clm( k )s true. So, Clm() stsfes ll the codtos of the prcple of Mthemtcl Iducto. Therefore, by the Prcple of Mthemtcl Iducto, Clm() s true for ll. WUCT Numbers 49

7 Exmple: Returg to our frst exmple, we hd Clm(): >.. From erler work, Clm() s true.. Assume Clm(k) s true for k, tht s, k > k (). Prove Clm( k ) s true, tht s, prove tht ( ) > ( k ) k. Note: RHS LHS k ( k ) k k k. k > k k k k k 5 > k RHS Therefore, ( ) > ( k ) s true. ( by hypothess ()) ( sce k ) k d so Clm( k ) Therefore, by the Prcple of Mthemtcl Iducto, Clm() s true for ll. WUCT Numbers 50

8 I summry: To estblsh fte fmly of clms, (Clm(), Clm(), Clm(), ) usg the Prcple of Mthemtcl Iducto, t s suffcet to crry out the followg two steps: Step : Prove tht the frst clm, Clm() s true. Step : Gve geerl proof tht for ech k., f Clm(k) s true, the Clm( k ) s lso true. Note: Logc otto c be used ltertvely to the word Clm. Tht s: To estblsh fte fmly of predctes, (P(), P(), P(), ) usg the Prcple of Mthemtcl Iducto, t s suffcet to crry out the followg two steps: Step : Prove tht the frst predcte, P() s true. Step : Gve geerl proof tht for ech k., f P(k) s true, the P( k ) s lso true. WUCT Numbers 5

9 Asde: Notes o Dvsblty The expresso b reds dvdes b, or, s fctor of b, or, s dvsor of b. It mes b s dvsble by, or b s multple of or more formlly, there s l. so tht b l. Exmples: 6 sce 6,. 7 ( ) l., 7l 6 F 5 reds 6 does ot dvde 5. Whe proof volves showg b, you eed to express b s multple of, tht s, fd l. so tht b l. Exmple: ( k ) To prove 4 k l. so tht 4 l., you eed to fd expresso for WUCT Numbers 5

10 Exercse: ( ) Prove tht 6 for ll. Let P() be. Step : P() s. Step : Assume P(k) s true for k, so tht tht s,.e Prove P( k ) s true, tht s, prove tht We eed to fd expresso for. m (usg l) Therefore, d so P( k ) s true. Therefore, by the PMI, P() s true for ll. WUCT Numbers 5

11 Asde: Notes o Fctorl The expresso! reds fctorl d s defed by ( )( )! K. Exmple: 5! Exercse: Prove ( ) Let P() be,!.. P() s:. Assume P(k) s true for k, Prove P( k ) WUCT Numbers 54

12 WUCT Numbers 55

13 WUCT Numbers 56.. Sgm otto The otto deotes the sum of the umbers K,,, ; tht s, K. Exmples: ( ) ( ) ( ) ( ) We wll use sgm otto to shorte expressos such s: ( ) ( ) K.

14 WUCT Numbers 57 It s mportt tht you stsfy yourself tht the followg equltes hold. () () () j j Also, t s esy to prove the followg sttemets: () ( ) m b b m m m, () m k k m m, Exercse: True or Flse? 9.