DISCRETE COMPUTATIONAL STRUCTURES

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DISCRETE COMPUTATIONAL STRUCTURES

• CS 23022
• Fall 2005

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CS 23022 OUTLINE

• Matrices Closures
• Counting Principles
• Discrete Probability
• Congruences
• Recurrence Relations
• Algorithm Complexity
• Graph Theory
• Trees Networks
• Grammars Languages

1. Sets
2. Logic
3. Proof Techniques
4. Algorithms
5. Integers Induction
6. Relations Posets
7. Functions
8. Boolean Algebra Combinatorial Circuits

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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 Algebra
Let 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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Minterm
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Maxterm
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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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CS 23022 OUTLINE

• Matrices Closures
• Congruences
• Counting Principles
• Discrete Probability
• Recurrence Relations
• Algorithm Complexity
• Graph Theory
• Trees Networks
• Grammars Languages

1. Sets
2. Logic
3. Proof Techniques
4. Algorithms
5. Integers Induction
6. Relations Posets
7. Functions
8. Boolean Algebra Combinatorial Circuits

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Learning Objectives

• Learn about matrices and their relationship with
  relations
• Become familiar with Boolean matrices
• Learn the relationship between Boolean matrices
  and different closures of a relation
• Explore how to find the transitive closure using
  Warshalls algorithm

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Matrices
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Matrices
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Matrices terms equal , square
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Matrices- terms zero matrix, diagonal elements
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Matrices- terms diagonal matrix, identity matrix
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Matrices Matrix Sum

• Two matrices are added only if they have the same
  number of rows and the same number of columns
• To determine the sum of two matrices, their
  corresponding elements are added

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Matrices Matrix Addition Example
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Matrices- Multiply a Constant x Matrix
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Matrices Matrix Difference
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Matrices - Properties
Commutative and Associative properties of Matrix
addition Distributive property of multiplication
over addition ( subtraction )-only holds for a
constant time a matrix sum (difference)
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Matrices

• The multiplication AB of matrices A and B is
  defined only if the number of columns of A is the
  same as the number of rows of B

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Matrices
Figure 4.1

• Let A aijmn be an m n matrix and B bjk
  np be an n p matrix. Then AB is defined
• To determine the (i, k)th element of AB, take the
  ith row of A and the kth column of B, multiply
  the corresponding elements, and add the result
• Multiply corresponding elements as in Figure 4.1

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Matrices
Note that the dimensions of AB are m x p.Then
(AB) x C is defined and has dimensions m x
qConvince yourself that A x (BC) is defined and
also has dimensions m x q
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Matrices Matrix transpose

• The rows of A are the columns of AT and the
  columns of A are the rows of AT

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Matrices - Symmetric
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Matrices

• Boolean (Zero-One) Matrices
• Matrices whose entries are 0 or 1
• Allows for representation of matrices in a
  convenient way in computer memory and for the
  design and implementation of algorithms to
  determine the transitive closure of a relation

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Matrices

• Boolean (Zero-One) Matrices
• The set 0, 1 is a lattice under the usual less
  than or equal to relation, where for all a, b ?
  0, 1, a ? b maxa, b and a ? b mina, b

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Matrices Logical Operations
Note join is the OR operation meet is the AND
operation
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Matrices
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Matrices Boolean Product
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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• ALGORITHM 4.3 Compute the transitive closure
• Input M Boolean matrices of the relation R
  npositive integers such
  that n n specifies the size of M
• Output T an n n Boolean matrix such that T is
  the transitive closure of M
• 1. procedure transitiveClosure(M,T,n)
• 2. begin
• 3. A M
• 4. T M
• 5. for i 2 to n do
• 6. begin
• 7. A //A Mi
• 8. T T ? A //T M ? M2 ?
  ? Mi
• 9. end
• 10. end

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Warshalls Algorithm for Determining the
Transitive Closure

• Previously, the transitive closure of a relation
  R was found by computing the matrices
  and then taking the Boolean join
• This method is expensive in terms of computer
  time
• Warshalls algorithm an efficient algorithm to
  determine the transitive closure

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Warshalls Algorithm for Determining the
Transitive Closure

• Let A a1, a2, . . . , an be a finite set, n
  1, and let R be a relation on A.
• Warshalls algorithm determines the transitive
  closure by constructing a sequence of n Boolean
  matrices

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Warshalls Algorithm for Determining the
Transitive Closure
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Warshalls Algorithm for Determining the
Transitive Closure
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Warshalls Algorithm for Determining the
Transitive Closure
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Warshalls Algorithm for Determining the
Transitive Closure

• ALGORITHM 4.4 Warshalls Algorithm
• Input M Boolean matrices of the relation R
• npositive integers such that n n specifies the
  size of M
• Output W an n n Boolean matrix such that W is
  the transitive closure of M
• 1. procedure WarshallAlgorithm(M,W,n)
• 2. begin
• 3. W M
• 4. for k 1 to n do
• 5. for i 1 to n do
• 6. for j 1 to n do
• 7. if Wi,j 1 then
• 8. if Wi,k 1 and Wk,j
  1 then
• 9. Wi,j 1
• 10. end

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Learning Objectives

• Learn the basic counting principlesmultiplication
  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. Morgans 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

• Consider 10 chips of 3 types ( R, W, B) with 5 R,
  3 W and 2 B
• Then, n 10, k 3, and n15, n23 and n3 2.
• The number of different arrangements of these 10
  chips is
• C(10,5) C(10-5,3) C(10-5-3,2) C(10,5)
  C(5,3) C(2,2)
• 10! / 5!(5!) 5!/ 3!(2!) 2! / 2!(1!) 10!
  / 5!3!2! n!/ n1!n2!n3!
• 10 9 8 7 6 / (3 2 1 2 1) 5
  9 8 7 2520

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Generalized Permutations and Combinations
Consider an 8-bit string. How many 8-bit strings
contain exactly three 1s ? Using the formula
above, with n 8 and k 3, the answer is
C(8,3). C(8,3) 8! / 3!(8-3)! 8!/ 3!5! 8
7 6 / 3 2 1 56 Examples 11100000
00000111 00011100 01010100 etc
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Generalized Permutations and Combinations
Suppose we have x y 3, with x 0, y 0. Then
n 3, k2 and the number of integer solutions
is C(32-1,2-1) C(4,1) 4 (0,3) , (1,2),
(2,1) , (3,0)
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Generalized Permutations and Combinations
Let objects 1,2,3,4,5, n 5, r3. Then the
number of 3-combinations of these objects ( with
repetitions allowed) is C(5-13,3) C(7,3) 7!
/ 3!(4!) 35
111, 222, 333, 444, 555 112, 113,114, 115 221,
223, 224 , 225 331, 332, 334, 335 441, 442, 443,
445 551, 552, 553, 554 123, 124, 125 , 134,
135,145 234, 235, 245 345
Repeat all 3 5 combinations Repeat two of
three 20 combinations No repeats 10
combinations
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Permutations and Combinations

• Permutations and Combinations - Rosen