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CSE 321 Discrete Structures

In this lecture on discrete structures, we delve into various proof techniques fundamental to mathematical reasoning. Key topics include direct proofs, contrapositive proofs, proof by contradiction, and equivalence proofs. We illustrate these methods with examples such as proving properties of odd and even integers and exploring combinatorial problems like the Game of Chomp and tiling challenges. Additionally, we highlight relevant homework assignments and university holidays, ensuring a comprehensive understanding of the material.

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CSE 321 Discrete Structures

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  1. CSE 321 Discrete Structures Winter 2008 Lecture 6 Proofs

  2. Announcements • Reading for this week • Today: 1.6, 1.7 • Homework 2 • Due Wednesday, January 23 • Martin Luther King Jr. Day, Mon., Jan 21 •  x (UniversityHoliday(x)  NoClass(x)) • UniversityHoliday(Monday)

  3. Highlights from Lecture 5 • Formal Reasoning • Build a proof, starting from hypotheses by applying rules of inference

  4. Proofs • Proof methods • Direct proof • Contrapositive proof • Proof by contradiction • Proof by equivalence

  5. Direct Proof • If n is odd, then n2 is odd Definition n is even if n = 2k for some integer k n is odd if n = 2k+1 for some integer k

  6. Contrapositive • Sometimes it is easier to prove qp than it is to prove pq • Prove that if ab n then an1/2 or bn1/2

  7. Proof by contradiction • Suppose we want to prove p is true. • Assume p is false, and derive a contradiction

  8. Contradiction example • Show that at least four of any 22 days must fall on the same day of the week

  9. Equivalence Proof • To show p1p2p3, we show p1p2, p2p3, and p3p1 • Show that the following are equivalent • p1: n is even • p2: n-1 is odd • p3: n2 is even

  10. The Game of Chomp

  11. Theorem: The first player can always win in an nm game • Every position is a forced win for player A or player B (this fact will be used without proof) • Any finite length, deterministic game with no ties is a win for player A or player B under optimal play

  12. Proof • Consider taking the lower right cell • If this is a forced win for A, then done • Otherwise, B has a move m that is a forced win for B, so if A started with this move, A would have a forced win

  13. Tiling problems • Can an nn checkerboard be tiled with 2  1 tiles?

  14. 8 8 Checkerboard with two corners removed • Can an 8  8 checkerboard with upper left and lower right corners removed be tiled with 2  1 tiles?

  15. 8 8 Checkerboard with one corner removed • Can an 8  8 checkerboard with one corner removed be tiled with 3  1 tiles?

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