MTH 231 Lecture Notes - Lecture 13: Mathematical Induction

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orchidpig108

20 Jul 2018

School

Oregon State University

Department

Mathematics

Course

MTH 231

Professor

All

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Proofs by mathematical induction: (cid:1827) =(cid:1827, (cid:1827)(cid:1515)(cid:1828)=(cid:1828)(cid:1515)(cid:1827) , (cid:1827)(cid:1514)(cid:1828)=(cid:1828)(cid:1514)(cid:1827, (cid:1827)(cid:1515)(cid:4666)(cid:1828)(cid:1514)(cid:1855)(cid:4667)= (cid:4666)(cid:1827)(cid:1515)(cid:1828)(cid:4667)(cid:1514)(cid:4666)(cid:1827)(cid:1515)(cid:1829)(cid:4667) (cid:1827)(cid:1514)(cid:4666)(cid:1828)(cid:1515)(cid:1855)(cid:4667)= (cid:4666)(cid:1827)(cid:1514)(cid:1828)(cid:4667)(cid:1515)(cid:4666)(cid:1827)(cid:1514)(cid:1829)(cid:4667) = (cid:1827) (cid:1514)(cid:1828) (cid:1830)(cid:1857) (cid:1839)(cid:1867)(cid:1870)(cid:1859)(cid:1866) (cid:1871) (cid:1838)(cid:1875: (cid:1827)(cid:1515)(cid:1828) =(cid:2869) <=> ,(cid:883)(cid:3409)(cid:3409)(cid:1866) (cid:1866)(cid:1856) (cid:1876) (cid:1827) (cid:1876) (cid:1827) (cid:1876) (cid:1827)<=> (cid:3410)(cid:883) (cid:1866)(cid:1856) (cid:1876) (cid:1827) Section 5. 1: method principle of mathematical introduction situation (p. m. i) Strategy: p (0), p (cid:894)(cid:1005)(cid:895), p (cid:894)(cid:1006)(cid:895), p (cid:894)(cid:1007)(cid:895) . Situation: want to prove n = {(cid:1004),(cid:1005),(cid:1006) } (cid:1866) (cid:4666)(cid:1866)(cid:4667)(cid:1871) (cid:1872)(cid:1870)(cid:1873)(cid:1857: show [(cid:4666)(cid:1866)(cid:4667) (cid:4666)(cid:1866)+(cid:883)(cid:4667)] is true, where (cid:4666)(cid:1866)(cid:4667)(cid:1871) (cid:1868) (cid:1866)(cid:1856) (cid:4666)(cid:1866)+(cid:883)(cid:4667)(cid:1871) (cid:1869) Method principle of mathematical introduction (p. m. i): show p (0) is true. =(cid:2868: base case: n=0, proof by p. m. i, proof p. m. i, (cid:862)base core(cid:863) how p (cid:894)(cid:1004)(cid:895) is true, (cid:862)induction hypothesis(cid:863) (i. h. ): suppose p(n) is true: show p(n+1) is true follows from supposing. (cid:4666)(cid:1866)(cid:4667) (cid:1875)(cid:1871) (cid:1872)(cid:1870)(cid:1873)(cid:1857). (cid:1877)(cid:1867)(cid:1873) (cid:1871) (cid:1867)(cid:1873)(cid:1864)(cid:1856) (cid:1871)(cid:1877) (cid:1875) (cid:1857)(cid:1866) (cid:1873)(cid:1871)(cid:1857) (cid:1872) (cid:1857) (cid:1835). (cid:1834: conclude (cid:1866)(cid:3410)(cid:882) (cid:4666)(cid:1866)(cid:4667)(cid:1871) (cid:1872)(cid:1870)(cid:1873)(cid:1857) Ex: show (cid:1866)(cid:3410)(cid:882) (cid:1863)= (cid:4666)+(cid:2869)(cid:4667)(cid:2870) (cid:1863)= (cid:4666)+(cid:2869)(cid:4667)(cid:2870) =(cid:2868: (cid:862)i. h. (cid:863): suppose (cid:1863)= (cid:4666)+(cid:2869)(cid:4667)(cid:2870) (cid:1858)(cid:1867)(cid:1870) (cid:1871)(cid:1867)(cid:1865)(cid:1857) (cid:1866) (cid:3410)(cid:882)

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MTH 231 Lecture Notes - Lecture 13: Mathematical Induction

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