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Induction

Induction. http://www.picshag.com/recursive-painting.html. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois. Last lecture: graphs and 2-way bounds. Terminology for graph connectivity

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Induction

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  1. Induction http://www.picshag.com/recursive-painting.html Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois

  2. 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

  3. Two-way bounding: set equality Claim: For any integer , is equal to .

  4. This lecture (and next): Induction • What is induction • Examples

  5. Does domino n fall?

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

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

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

  9. Simple math example Claim: for all natural integers .

  10. Basic structure of induction proof Claim: Inductive step: Base case: is true. Weak Induction Inductive conclusion Inductive hypothesis Strong Induction Inductive conclusion Inductive hypothesis

  11. Another math example Claim: For ,

  12. Number theory example Claim: For any natural integer , is divisible by .

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

  14. 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

  15. Next class • Induction with graphs, stamps, and games

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