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Unformatted text preview: LECTURE 15: INDUCTION STRENGTHENED ( 6.4) I think I should have no other mortal wants, if I could always have plenty of music. It seems to infuse strength into my limbs and ideas into my brain. Life seems to go on without effort, when I am filled with music. George Eliot (1819 - 1880) 1. Strong Induction Recall our infinite staircase. To climb it, we needed to do two things. First, climb the bottom step. Second, use the fact that we climbed the k th step to climb the ( k + 1)st step. But we are not using everything available to us. We didnt only climb the k th step, we also climbed all the previous steps. So we can strengthen our inductive case. We do this by not only assuming P ( k ); but rather we assume P ( i ) for all i in the range 1 i k . We still show that we can then get P ( k +1). Why dont we always do this strong induction? Because sometimes it just complicates things....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11