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DISCRETE COMPUTATIONAL STRUCTURES

DISCRETE COMPUTATIONAL STRUCTURES. CSE 2353 Fall 2005. CSE 2353 OUTLINE. Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra. Learning Objectives. Learn the basic counting principles—multiplication and addition

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DISCRETE COMPUTATIONAL STRUCTURES

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  1. DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Fall 2005

  2. CSE 2353 OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

  3. Learning Objectives • Learn the basic counting principles—multiplication and addition • Explore the pigeonhole principle • Learn about permutations • Learn about combinations Discrete Mathematical Structures: Theory and Applications

  4. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

  5. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

  6. Basic Counting Principles • There are three boxes containing books. The first box contains 15 mathematics books by different authors, the second box contains 12 chemistry books by different authors, and the third box contains 10 computer science books by different authors. • A student wants to take a book from one of the three boxes. In how many ways can the student do this? Discrete Mathematical Structures: Theory and Applications

  7. Basic Counting Principles • Suppose tasks T1, T2, and T3 are as follows: • T1: Choose a mathematics book. • T2: Choose a chemistry book. • T3: Choose a computer science book. • Then tasks T1, T2, and T3 can be done in 15, 12, and 10 ways, respectively. • All of these tasks are independent of each other. Hence, the number of ways to do one of these tasks is 15 + 12 + 10 = 37. Discrete Mathematical Structures: Theory and Applications

  8. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

  9. Basic Counting Principles • Morgan is a lead actor in a new movie. She needs to shoot a scene in the morning in studio A and an afternoon scene in studio C. She looks at the map and finds that there is no direct route from studio A to studio C. Studio B is located between studios A and C. Morgan’s friends Brad and Jennifer are shooting a movie in studio B. There are three roads, say A1, A2, and A3, from studio A to studio B and four roads, say B1, B2, B3, and B4, from studio B to studio C. In how many ways can Morgan go from studio A to studio C and have lunch with Brad and Jennifer at Studio B? Discrete Mathematical Structures: Theory and Applications

  10. Basic Counting Principles • There are 3 ways to go from studio A to studio B and 4 ways to go from studio B to studio C. • The number of ways to go from studio A to studio C via studio B is 3 * 4 = 12. Discrete Mathematical Structures: Theory and Applications

  11. Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

  12. Basic Counting Principles • Consider two finite sets, X1and X2. Then • This is called the inclusion-exclusion principle for two finite sets. • Consider three finite sets, A, B, and C. Then • This is called the inclusion-exclusion principle for three finite sets. Discrete Mathematical Structures: Theory and Applications

  13. Pigeonhole Principle • The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. Discrete Mathematical Structures: Theory and Applications

  14. Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

  15. Discrete Mathematical Structures: Theory and Applications

  16. Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

  17. Permutations Discrete Mathematical Structures: Theory and Applications

  18. Permutations Discrete Mathematical Structures: Theory and Applications

  19. Combinations Discrete Mathematical Structures: Theory and Applications

  20. Combinations Discrete Mathematical Structures: Theory and Applications

  21. Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

  22. Generalized Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

  23. CSE 2353 OUTLINE Sets Logic Proof Techniques Integers and Induction Relations and Posets Functions Counting Principles Boolean Algebra

  24. Learning Objectives • Learn about Boolean expressions • Become aware of the basic properties of Boolean algebra • Explore the application of Boolean algebra in the design of electronic circuits • Learn the application of Boolean algebra in switching circuits Discrete Mathematical Structures: Theory and Applications

  25. Two-Element Boolean Algebra Let B = {0, 1}. Discrete Mathematical Structures: Theory and Applications

  26. Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

  27. Discrete Mathematical Structures: Theory and Applications

  28. Discrete Mathematical Structures: Theory and Applications

  29. Discrete Mathematical Structures: Theory and Applications

  30. Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

  31. Two-Element Boolean Algebra Discrete Mathematical Structures: Theory and Applications

  32. Discrete Mathematical Structures: Theory and Applications

  33. Discrete Mathematical Structures: Theory and Applications

  34. Discrete Mathematical Structures: Theory and Applications

  35. Discrete Mathematical Structures: Theory and Applications

  36. Discrete Mathematical Structures: Theory and Applications

  37. Discrete Mathematical Structures: Theory and Applications

  38. Discrete Mathematical Structures: Theory and Applications

  39. Boolean Algebra Discrete Mathematical Structures: Theory and Applications

  40. Boolean Algebra Discrete Mathematical Structures: Theory and Applications

  41. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  42. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  43. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  44. Logical Gates and Combinatorial Circuits Discrete Mathematical Structures: Theory and Applications

  45. Discrete Mathematical Structures: Theory and Applications

  46. Discrete Mathematical Structures: Theory and Applications

  47. Discrete Mathematical Structures: Theory and Applications

  48. Discrete Mathematical Structures: Theory and Applications

  49. Discrete Mathematical Structures: Theory and Applications

  50. Discrete Mathematical Structures: Theory and Applications

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