DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Summer 2005

CSE 2353 OUTLINE 1.Sets 2.Logic 3.Proof Techniques 4.Integers and Induction 5.Relations and Posets 6.Functions 7.Counting Principles 8.Boolean Algebra

Discrete Mathematical Structures: Theory and Applications 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 T 1, T 2, and T 3 are as follows:  T 1 : Choose a mathematics book.  T 2 : Choose a chemistry book.  T 3 : Choose a computer science book.  Then tasks T 1, T 2, and T 3 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 = 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 A 1, A 2, and A 3, from studio A to studio B and four roads, say B 1, B 2, B 3, and B 4, 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, X 1 and X 2. 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

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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

CSE 2353 OUTLINE 1.Sets 2.Logic 3.Proof Techniques 4.Integers and Induction 5.Relations and Posets 6.Functions 7.Counting Principles 8.Boolean Algebra

Discrete Mathematical Structures: Theory and Applications 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

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

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

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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

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Discrete Mathematical Structures: Theory and Applications 61 Logical Gates and Combinatorial Circuits  The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression.

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