1 csc-2259 discrete structures konstantin busch louisiana state university k. busch - lsu

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CSC-2259 Discrete Structures

Konstantin Busch

Louisiana State University

K. Busch - LSU

Topics to be covered

• Logic and Proofs• Sets, Functions, Sequences, Sums• Integers, Matrices• Induction, Recursion• Counting• Discrete Probability• Graphs

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Binary Arithmetic

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Decimal Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Binary Digits: 0, 1

Numbers: 9, 28, 211, etc

Numbers: 1001, 11100, 11010011, etc

(also known as bits)

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Binary

1001

91821202021 0123

Decimal

9

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Binary Addition Binary Multiplication

1001 (9)+ 1 1 (3)------1100 (12)

1001 (9) x 1 1 (3) ------ 1001+ 1001--------- 11011 (27)

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x

yz

xy

z x zGates

AND OR NOT

x y z

0 0 0

0 1 0

1 0 0

1 1 1

AND

Binary Logic

x y z

0 0 0

0 1 1

1 0 1

1 1 1

OR

x z

0 1

1 0

NOT

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An arbitrary binary function is implementedwith NOT, AND, and OR gates yxxxf n ),,,( 21

y

1x

2x

2x

nx…

NOT AND

OR

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Propositional Logic

Proposition: a declarative sentence which is either True or False

Examples:Today is Wednesday (False)Today it Snows (False)1+1 = 2 (True)1+1 = 1 (False)H20 = water (True)

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Propositions can be combined usingthe binary operators AND, OR, NOT

We can map to binary values:True = 1False = 0

))(()( cbaqp Example:

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True True True

True False False

False True True

False False True

x y yxImplication

x implies y

“You get a computer science degree only if you are a computer science major”

You get a computer science degree:x:y You are a computer science major

yx

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True True True

True False False

False True False

False False True

x y yxBi-conditional

x if and only if y

“There is a received phone call if and only if there is a phone ring”

:x:yThere is a received phone call

There is a phone ring

yx

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Sets

Set is a collection of elements:

Real numbers R

Integers Z

Empty Set

Students in this room

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Subset }5,4,3,2,1{}4,2{

Basic Set Operations

2

4

1

3

5

Union }5,4,3,2,1{}5,4,2{}3,2,1{

241

3

5

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Intersection }2{}5,4,2{}3,2,1{

241

3

5

Complement }5,3,2{}4,1{

}5,4,3,2,1{universeK. Busch - LSU

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DeMorgan’s Laws

BABA

BABA

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Inclusion-Exclusion

A B

C

|CBA|

||||||

|||||| ||

CBCABA

CBACBA

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Powersets

Contains all subsets of a set

}}3,2,1{},3,1{},3,2{},2,1{},3{},2{},1{,{Q

}3,2,1{A

||2|| AQ

Powerset of A

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Counting

Suppose we are given four objects: a, b, c, d

How many ways are there to order the objects? 4321!4

a,b,c,db,a,c,da,b,d,cb,a,d,c … and so on

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CombinationsGiven a set S with n elementshow many subsets exist with m elements?

)!(!

!

mnm

n

m

n

Example: 3)!23(!2

!3

2

3

}3,1{},3,2{},2,1{ }3,2,1{SK. Busch - LSU

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Sterling’s Approximation

n

e

nnn

2!

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Probabilities

What is the probability the a dice gives 5?

Sample space = {1,2,3,4,5,6}

Event set = {5}

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space sample of Size

setevent of Size{5})obability(Pr

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What is the probability that two dice give the same number?

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Sample Space = {{1,1},{1,2},{1,3}, …., {6,5}, {6,6}}

Event set = {{1,1},{2,2},{3,3},{4,4},{5,5},{6,6}}

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6

space sample of Size

setevent of Sizenumber}) {sameobability(Pr

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Randomized Algorithms

Quicksort(A):If ( |A| == 1)

return the one item in AElse p = RandomElement(A) L = elements less than p H = elements higher than p B = Quicksort(L) C = Quicksort(H) return(BC)

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Graph Theory

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Miami

Atlanta

New York

BostonChicago

Baton Rouge

Las Vegas

San Francisco

Los Angeles

2000 miles1500 miles

1500

15001000

2000

700

1500

300

800

700

1500

1000

1000

800

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Shortest Path from Los Angeles to Boston

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Boston

Los Angeles

20001500

1500

15001000

2000

700

1500

300

800

700

1500

1000

1000

800

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Maximum number of edges in a graphwith nodes:

22

)1(

)!2(!2

!

2

2 nnnn

n

nn

Clique with five nodes

5n10edges

n

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Other interesting graphs

Bipartite Graph

Trees

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Recursion

)1()( nfnnf

1)1(2)( nfnf

Basis1)1( f

1)1( f

Basis

nnnf )1(4321)( Sum of arithmetic sequence

Sum of geometric sequence)1(3210 22222)( nnf

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nn

fnf

2

2)(

)2(1)( nfnfnf

1)1( f

1)1(,0)0( ff

Fibonacci numbers

Basis

Divide and conquer algorithms (Quicksort)

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Induction

Contradiction

Pigeonhole principle

Proof Techniques

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2)1(

)(

nn

nf

Proof by Induction

Induction Basis:2

)11(11)1(

f

Induction Hypothesis:2

)1()1(

nnnf

2)1(

22)1(

)1()(2

nnnnnn

nnfnnf

Induction Step:

)1()( nfnnf Prove:

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Proof by Contradiction

2 is irrational

Supposen

m2 ( and have no common

divisor greater than 1 )

m n

2

2

2n

m m2 is even m is even

2 n2 = 4k2 n2 = 2k2 n is even

m=2k

ContradictionK. Busch - LSU