1

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

K. Busch - LSU 2

Binary Arithmetic

3

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)

K. Busch - LSU

4

Binary

1001

91821202021 0123

Decimal

9

K. Busch - LSU

5

Binary Addition Binary Multiplication

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

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

K. Busch - LSU

6

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

K. Busch - LSU

7

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

y

1x

2x

2x

nx…

NOT AND

OR

K. Busch - LSU

8

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)

K. Busch - LSU

9

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

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

))(()( cbaqp Example:

K. Busch - LSU

10

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

K. Busch - LSU

11

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

K. Busch - LSU

12

Sets

Set is a collection of elements:

Real numbers R

Integers Z

Empty Set

Students in this room

K. Busch - LSU

13

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

K. Busch - LSU

14

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

15

DeMorgan’s Laws

BABA

BABA

K. Busch - LSU

16

Inclusion-Exclusion

A B

C

|CBA|

||||||

|||||| ||

CBCABA

CBACBA

K. Busch - LSU

17

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

K. Busch - LSU

18

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

K. Busch - LSU

19

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

20

Sterling’s Approximation

n

e

nnn

2!

K. Busch - LSU

21

Probabilities

What is the probability the a dice gives 5?

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

Event set = {5}

K. Busch - LSU

6

1

space sample of Size

setevent of Size{5})obability(Pr

22

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

22

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}}

K. Busch - LSU

36

6

space sample of Size

setevent of Sizenumber}) {sameobability(Pr

23

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)

K. Busch - LSU

24

Graph Theory

K. Busch - LSU

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

25

Shortest Path from Los Angeles to Boston

K. Busch - LSU

Boston

Los Angeles

20001500

1500

15001000

2000

700

1500

300

800

700

1500

1000

1000

800

26

Maximum number of edges in a graphwith nodes:

22

)1(

)!2(!2

!

2

2 nnnn

n

nn

Clique with five nodes

5n10edges

n

K. Busch - LSU

27

Other interesting graphs

Bipartite Graph

Trees

K. Busch - LSU

28

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

K. Busch - LSU

29

nn

fnf

2

2)(

)2(1)( nfnfnf

1)1( f

1)1(,0)0( ff

Fibonacci numbers

Basis

Divide and conquer algorithms (Quicksort)

K. Busch - LSU

30

Induction

Contradiction

Pigeonhole principle

Proof Techniques

K. Busch - LSU

31

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:

K. Busch - LSU

32

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