Download Introduction - LSU Computer Science

CSC-2259 Discrete Structures
Konstantin Busch
Louisiana State University
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Topics to be covered
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Logic and Proofs
Sets, Functions, Sequences, Sums
Integers, Matrices
Induction, Recursion
Counting
Discrete Probability
Graphs
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Binary Arithmetic
Decimal Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Numbers:
9, 28, 211, etc
Binary Digits: 0, 1
(also known as bits)
Numbers: 1001, 11100, 11010011, etc
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Binary
Decimal
1001
9
1 2  0  2  0  2  1 2  8  1  9
3
2
1
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0
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Binary Addition
1001 (9)
+ 1 1 (3)
-----1100 (12)
Binary Multiplication
1001 (9)
x 1 1 (3)
-----1001
+ 1001
--------11011
(27)
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Binary Logic
AND
Gates
OR
x
y
z
x
y
z
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0
1
0
1
1
1
1
1
1
1
x
y
z
AND
x
y
NOT
z
OR
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x
z
0
1
1
0
x
z
NOT
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An arbitrary binary function is implemented
with NOT, AND, and OR gates
f ( x1 , x2 ,, xn )  y
x1
x2
x2
y
…
OR
xn
NOT
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Propositional Logic
Proposition: a declarative sentence which
is either True or False
Examples: Today is Wednesday
Today it Snows
1+1 = 2
1+1 = 1
H20 = water
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(False)
(False)
(True)
(False)
(True)
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We can map to binary values: True = 1
False = 0
Propositions can be combined using
the binary operators AND, OR, NOT

Example:


( p  q)  (a  (b  c))
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Implication
y
x y
x
True
True
True
True
False
False
False
True
True
False
False
True
x implies y
x y
“You get a computer science degree
only if you are a computer science major”
x : You get a computer science degree
y : You are a computer science major
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Bi-conditional
y
x y
x
True
True
True
True
False
False
False
True
False
False
False
True
x if and only if y
x y
“There is a received phone call if and only if
there is a phone ring”
There is a received phone call
y : There is a phone ring
x:
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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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Basic Set Operations
Subset {2,4}  {1,2,3,4,5}
1
3
2
4 5
Union {1,2,3}  {2,4,5}  {1,2,3,4,5}
3
1
2
4
5
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{1,2,3}  {2,4,5}  {2}
Intersection
3
1
2
4
5
Complement
{1,4}  {2,3,5}
universe {1,2,3,4,5}
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DeMorgan’s Laws
A B  A B
A B  A B
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Inclusion-Exclusion
| A B C |  | A|  | B |  | C |
 | A B |  | AC |  | B C |
 | A BC|
A
B
C
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Powersets
Contains all subsets of a set
A  {1,2,3}
Powerset of A
Q  {,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}
| Q | 2
| 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?
4! 1 2  3  4
a,b,c,d
b,a,c,d
a,b,d,c
b,a,d,c … and so on
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Combinations
Given a set S with n elements
how many subsets exist with m elements?
n
n!
  
 m  m!(n  m)!
Example:
S  {1,2,3}
 3
3!
  
3
 2  2!(3  2)!
{1,2},{2,3},{1,3}
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Sterling’s Approximation
n
n!  2n  
e
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n
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Probabilities
What is the probability the a dice gives 5?
Event set = {5}
Sample space = {1,2,3,4,5,6}
Size of event set
1
Pr obability( {5}) 

Size of sample space 6
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What is the probability that two dice
give the same number?
Event set = {{1,1},{2,2},{3,3},{4,4},{5,5},{6,6}}
Sample Space = {{1,1},{1,2},{1,3}, …., {6,5}, {6,6}}
Size of event set
6
Pr obability( {same number}) 

Size of sample space 36
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Randomized Algorithms
Quicksort(A):
If ( |A| == 1)
return the one item in A
Else
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
San Francisco
800
700
2000 miles
1500 miles
300
1500
Las Vegas
1500
Chicago
1000
1500
1000
1000
2000
Los Angeles
Boston
New York
800
Atlanta
700
Baton Rouge
1500
Miami
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Shortest Path from Los Angeles to Boston
1500
2000
800
1500
1500
1000
700
1500
1000
300
Boston
800
1000
700
2000
1500
Los Angeles
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Maximum number of edges in a graph
with n nodes:
2
n
 
n!
n(n  1) n  n
  


2
2
 2  2!(n  2)!
n5
edges  10
Clique with five nodes
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Other interesting graphs
Trees
Bipartite Graph
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Recursion
Sum of arithmetic sequence
f (n )  1  2  3  4    (n  1)  n
f (n )  n  f (n  1)
Basis
f (1)  1
Sum of geometric sequence
f (n )  20  21  22  23    2(n 1)
Basis
f (n )  2  f (n  1)  1
f (1)  1
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Fibonacci numbers
Basis
f (n )  f n  1  f (n  2)
f (0)  0, f (1)  1
Divide and conquer algorithms (Quicksort)
n

f (n )  2  f    n
2
f (1)  1
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Proof Techniques
Induction
Contradiction
Pigeonhole principle
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Proof by Induction
f (n )  n  f (n  1)
n (n  1)
Prove: f (n ) 
2
1(1  1)
Induction Basis: f (1)  1 
2
Induction Hypothesis:
(n  1)n
f (n  1) 
2
Induction Step:
(n  1)n n 2  n n (n  1)
f (n )  n  f (n  1)  n 


2
2
2
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Proof by Contradiction
is irrational
2
Suppose
m2
2 2
n
2 n2 = 4k2
m
2
n
(
m and n have no common
divisor greater than 1 )
m2 is even
n2 = 2k2
m=2k
m is even
n is even
Contradiction
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