lec0228 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Siebel Center, rm 2213 Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 6 available. Due 03/05, 8a. Midterm 1 returned Thursday for sure. New section, Thursday 6-7p in Siebel 1129 has been CANCELLED!!!
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Cs173 - Spring 2004 CS173 Inductive Definitions Our examples so far have been inductively defined functions. Sets can be defined inductively, too. Recursive Cases Base Case Give an inductive definition of S = {x: x is a multiple of 3} 3 S x,y S x + y S x,y S x - y S No other numbers are in S. “I LOVE my definition. It’s perfect!” “Oh yeah? Prove it!”
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Cs173 - Spring 2004 CS173 Inductive Definitions We want to show that my definition of S: 3 S x,y S x + y S x,y S x - y S No other numbers are in S. Contains the same elements as the set: T = {x: x is a multiple of 3} To prove S = T, show T S S T
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Cs173 - Spring 2004 CS173 Inductive Definitions We start with T S. If x T, then x = 3k for some integer k. We show by induction on |k| that 3k S. Base Case (k = 0): 0 S since 3 S by rule 1, and 3 - 3 S by rule 3. Assume 3k, -3k S, show that 3(k+1), -3(k+1) S.
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Cs173 - Spring 2004 CS173 Inductive Definitions We start with T S. If x T, then x = 3k for some integer k. We show by induction on |k| that 3k S. Assume 3k, -3k S, show that 3(k+1), -3(k+1) S. 3k S by inductive hypothesis. 3 S by rule 1. 3k + 3 = 3(k+1) S by rule 2. 0 S by base case. 0 - 3(k+1) = -3(k+1) S by rule 3.
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Cs173 - Spring 2004 CS173 Inductive Definitions Next we show that S T. That is, if an number x is described by S, then it is a multiple of 3. Observe that by rule 4, all numbers in S are created by a finite number of applications of rules 1,2, and 3. We use the number of rule applications as our induction counter. For example: 3 S by 1 application of rule 1. 0 S by 2 rule applications (rules 1 and 3). 9 S by 3 applications (rule 1 once and rule 2 twice).
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Cs173 - Spring 2004 CS173 Inductive Definitions Next we show that S T. That is, if an number x is described by S, then it is a multiple of 3. Base Case (k=1): If x S by 1 rule application, then it must be rule 1 and x = 3, which is clearly a multiple of 3.
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Cs173 - Spring 2004 CS173 Inductive Definitions Next we show that S T. That is, if an number x is described by S, then it is a multiple of 3. Assume any number described by k or fewer applications of the rules in S is a multiple of 3 and prove that any number described by (k+1) applications of the rules is also a multiple of 3.
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lec0228 - CS 173: Discrete Mathematical Structures Cinda...

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