DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Summer 2005

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 Principles8. Boolean Algebra

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Learning Objectives

Learn the basic counting principles—multiplication and addition

Explore the pigeonhole principle

Learn about permutations

Learn about combinations

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Basic Counting Principles

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Basic Counting Principles

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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?

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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.

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Basic Counting Principles

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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?

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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.

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Basic Counting Principles

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Basic Counting Principles

Consider two finite sets, X1 and 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.

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Pigeonhole Principle

The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.

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Pigeonhole Principle

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Pigeonhole Principle

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Permutations

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Permutations

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Combinations

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Combinations

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Generalized Permutations and Combinations

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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

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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

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Two-Element Boolean AlgebraLet B = {0, 1}.

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Two-Element Boolean Algebra

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Two-Element Boolean Algebra

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Two-Element Boolean Algebra

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Boolean Algebra

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Boolean Algebra

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Logical Gates and Combinatorial Circuits

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Logical Gates and Combinatorial Circuits

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Logical Gates and Combinatorial Circuits

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Logical Gates and Combinatorial Circuits

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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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