13-Induction

# 13-Induction - Discussion#13 Induction(the process of deriving generalities from particulars Mathematical Induction(deductive reasoning over the

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Discussion #13 1/18 Discussion #13 Induction (the process of deriving generalities from particulars) Mathematical Induction (deductive reasoning over the natural numbers)

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Discussion #13 2/18 Topics Proof by Induction Summary of Proof Techniques
Discussion #13 3/18 Statements that Depend on a Natural Number Sometimes we wish to prove that a statement is true for all natural numbers n 0 or for all natural numbers greater than some initial number. Examples: Prove: The sum of the numbers 1 to n is n(n+1)/2. Prove: n 2 > 2n + 1, for n 3. Prove: If T is a full binary tree, then the number of nodes at each level n of T is 2 n . Prove: If E is an expression with n occurrences of binary operators, then E has n+1 operands. Each statement S to be proved depends on a single number n. We can write S(n) to denote the statement that depends on n. Thus, for the first statement above, S(n) is “The sum of the numbers 1 through n is n(n+1)/2.”

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Discussion #13 4/18 Prove S(n) for All n m To prove a statement S(n) for n m, we must show that S(m), S(m+1), S(m+2), … are all true. Example: For S(n) = “The sum of the numbers 1 through n is n(n+1)/2” we must prove: The sum of the numbers 1 to 1 is 1(1+1)/2. The sum of the numbers 1 to 2 is 2(2+1)/2. The sum of the numbers 1 to 3 is 3(3+1)/2. Unfortunately, we can never reach the end. The simple pattern of one after the next, however, lets us solve the problem.
Discussion #13 5/18 Modus Ponens is the Key S(1) S(1) S(2) S(2) S(2) S(3) S(3) S(k) S(k) S(k+1) S(k+1) If we can get started … And then show for an arbitrary natural number k that this implication is a tautology … We can conclude that S is true for the next natural number, k+1 (if it is true for k). Then since k is chosen arbitrarily (works for any natural number), S must be true for k = 2, 3, … (all natural numbers).

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Discussion #13 6/18 Induction Fundamentals by Example Prove: S(n), e.g. Prove: “The sum of the numbers from 1 to n is n(n+1)/2.” Basis: (i.e. this is how we get started) S(1) holds. i.e. The sum of the numbers from 1 to 1 is 1(1+1)/2 is a true statement.
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## This note was uploaded on 03/02/2012 for the course C S 236 taught by Professor Michaelgoodrich during the Winter '12 term at BYU.

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13-Induction - Discussion#13 Induction(the process of deriving generalities from particulars Mathematical Induction(deductive reasoning over the

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