Introduction to Discrete Math'

1
Introduction to Discrete Math.

• 2003. 9.
• ??????? ???????
• ? ? ?

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Discrete vs. Continuous

• Continuous
• Without interruption and without abrupt changes
• ???
• Analog watch
• A set of real numbers (Uncountable)
• Geometrically continuous representation
• Discrete
• Digital watch
• Works in discrete quantities
• A (finite) countable set of integers
• Geometrically disconnected elements
• Computer a finite discrete system

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Discrete vs. Continuous

• x x is a real number and 3.2 lt x lt9.5
• x x is an integer and 3.2 lt x lt 9.5
• Finite Countable Set
• x x is a positive integer
• Infinite Countable Set
• x x is a positive real number

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Discrete Mathematics

• Mathematics A tool for problem solving
• Mathematical modeling
• The methodology for using mathematics for problem
  solving
• Discrete Mathematics
• a framework to understand the problems of
  designing digital computer systems and programs

Problem
Solution to the problem
Abstract Model
Transformed Model
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Example-1
Jack has twelve apples, which cost him ten cents
apiece. How many apples must he sell at twenty
cents apiece before he begins to make a profit?
Problem
Income (sale price) (number sold) Expenses
(cost) (number bought)
Abstract Model
(sale price) x (cost) (number bought)
20 x 10 12
Transformed Model
Rules for simplifying the equation
x 120 / 20
Solution
x 6
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Example (problem)-2

• The following table contains distance mileage
  distances between 6 cities.
• Find a road network of minimal total length that
  connects all the cities.

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Example (Abstract Model)-2

• Graph
• Vertex a circle (city)
• Edge a line (road between two cites)

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Example (Rules for Transform)-2

• Goal to reduce the graph to another graph of
  minimal total length that connects all the
  cities.
• Algorithm (a set (sequence) of rules)
• Select any vertex (city)
• Connect it to the nearest adjacent vertex (city)
• If there are unconnected vertices, locate the
  nearest one to one of the connected vertices, and
  connect these two vertices
• Repeat the step 3 until all the vertices are
  connected

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Example (Transform and Solve)-2
20
C
5
B
5
A
F
20
10
10
E
D
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Example-3
C
3
B
2

• Exercise 1.1 (modified)

A
3
G
1
3
D
F
3
E
C
3
B
2
A
3
G
1
D
3
F
3
E
There are several minimal networks (not unique).
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Algorithm Description

• Alternatives
• Natural language (English, ??)
• Ambiguous
• Too long and too difficult to understand
• A specific computer programming language
• Too much details
• There is no universal language for all
  programmers
• Flowchart
• Not good for defining structured algorithms
• Too much freedom in describing control structures
• Pseudo-code
• A small set of structured language elements
• Unambiguous and independent on a specific
  programming language
• Very easy to translate the algorithm into an
  actual programming language

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Basic Elements of Pseudo-code

• Basic Building Block
• Statements
• Assignment statement
• Action statement
• Control statement

begin statements end.
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Assignment Statements

• A value to be assigned to a variable
• variable_name ? expression
• A ? A 1

Algorithm to compute the sum of two
numbers. begin Input First and Second Sum ?
First Second Output Sum end.
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Action Statements

• Describe an action
• Input First and Second
• Output Sum
• Connect u to the nearest connected vertex

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Control Statements

• Describe the flow of control through the
  algorithm
• Determine the execution order of the statements
  of the algorithm
• Sequence (or compound) statements
• Conditional Statements
• Iterative Statements

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Compound Statements

• A list of statements to be executed in the order
  given.
• Enclosed by begin and end

Statement 1
begin Statement1 Statement2 Statementn end.
Statement 2

Statement n
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Conditional Statements

• The selection of one alternative between two
• if condition then
• Statement

if condition then Statement1 else Statement2
Condition
true
false
Condition
false
true
Statement
Statement 2
Statement 1
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Example-Conditional Statements
Algorithm to compute the absolute value of
n. begin Input n if n lt 0 then n ? - n abs
? n Output n end.
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Example- if-else
Algorithm to find the larger value of two input
values. begin Input a and b if a gt b
then c ? a else c ? b Output c end.
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Iterative Statements

• for v ? initial_value to final_value do
• Statement
• for all elements of a set do
• Statement
• while expression do
• Statement
• repeat
• Statement1
• Statement2
• Statementn
• until condition

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Example-for
Algorithm to sum the 1st ten positive integers.
begin sum ? 0 for i ? 1 to 10 do sum ? sum
i Output sum end.
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Example-while

• 0 or more execution of the statements

Algorithm to sum the 1st n positive odd
integers. begin sum ? 0 i ? 1 Input
n while i lt 2n do begin sum ? sum
i i ? i 2 end Output sum end.
false
i lt 2n
true
sum ? sum i
i ? i 2
Output sum
End
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Example-repeat-until

• 1 or more execution

Algorithm to sum the 1st n positive odd
integers. begin sum ? 0 i ? 1 Input
n repeat sum ? sum i i ? i 2 until i gt
2n if n 0 then sum ? 0 Output sum end.
sum ? sum i
i ? i 2
false
i gt 2n
true
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Comments on Algorithms

• Correctness
• Efficiency

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HW 1

• Exercises 1.2
• 1, 2, 3
• Review Problems
• 1, 2, 3, 4, 5, 8, 9
• ???
• Ex1.2? 2
• R.P? 2, 4, 8 ?