Induction

Discrete Structures (CS 173)Madhusudan Parthasarathy, University of Illinois1

http://www.picshag.com/recursive-painting.html

Last lecture: graphs and 2-way bounds

• Terminology for graph connectivity– Walk, path, cycle, acyclic, closed, Euler circuit,

distance, diameter, connected components

• Graph coloring and how to apply it

• How to use two-way bounding in a variety of settings

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Two-way bounding: set equalityClaim: For any integer , is equal to .

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This lecture (and next): Induction

• What is induction

• Examples

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Does domino n fall?

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Does domino n fall?• Suppose domino k falls. Then domino k+1 falls.

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Does domino n fall?• Suppose domino k falls. Then domino k+1 falls.

• The first domino falls

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Induction

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Inductive hypothesis: Suppose domino k falls.Inductive conclusion: Domino k+1 falls.Base case: The first domino falls.

Simple math exampleClaim: for all natural integers .

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Basic structure of induction proofClaim: Inductive step:

Base case: is true.10

Inductive hypothesis

Inductive conclusion

Weak Induction

Strong Induction

Inductive hypothesis

Inductive conclusion

Another math exampleClaim: For ,

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Number theory exampleClaim: For any natural integer , is divisible by .

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Geometrical exampleClaim: For any positive integer , a checkerboard with one corner square removed can be tiled using right triminos.

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right trimino21×21

22×22

Things to remember

• Induction requires demonstrating a base case and an inductive step

• Inductive step usually involves showing that or – Typically, this requires writing in terms of

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Next class• Induction with graphs, stamps, and games

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