The LNM Institute of Information Technology
CSE 219: Discrete Mathematical Structures
Assignment 3
Q1. Let be a relation from a set
is the set of ordered pairs
pairs
to a set . The inverse relation from to , denoted by
,
The complementary relation is the
The LNM Institute of Information Technology
CSE 219: Discrete Mathematical Structures
Assignment 2
Q1. Find the domain and range of these functions.
a) the function that assigns to each pair of positive integers the first integer of the pair.
b) the funct
Mathematical Induction
Part One
Let P be some property. The principle of mathematical
induction states that if
If it starts
P(0) is true
true
and
and it stays
true
k . (P(k) P(k+1)
then
n . P(n)
then it's
always true.
Induction, Intuitively
It's true for
Introduction
Reformulation
The Main Theorem
Results & Conclusions
How Many Ways Can We Tile a Rectangular
Chessboard With Dominos?
Counting Tilings With Permanents and Determinants
Brendan W. Sullivan
Carnegie Mellon University
Undergraduate Math Club
Feb
The LNM Institute of Information Technology
CSE 219: Discrete Mathematical Structures
Assignment 1
Q1. Let p and q be the propositions
p : I bought a lottery ticket this week.
q : I won the million dollar jackpot on Friday.
Express each of these propositi
You Never Escape Your
Relations
1
Relations
If we want to describe a relationship between
elements of two sets A and B, we can use ordered
pairs with their first element taken from A and
their second element taken from B.
Since this is a relation between
Domino Tilings of the Chessboard
An Introduction to Sampling
and Counting
Dana Randall
Schools of
Computer Science
and Mathematics
Georgia Tech
Building short walls
How many ways are there to
build a 2 x n wall with 1 x 2 bricks?
2
n
1
2
Building short wa
Injections and Composition
Theorem: If f : A B is an injection and
g : B C is an injection, then the
function g f : A C is an injection.
Our goal will be to prove this result. To
do so, we're going to have to call back to
the formal definitions of injecti
Theorem: Let A, B, and C be any sets. Then
(A B) C = (A C) (B C)
C
A
B
Theorem: Let A, B, and C be any sets. Then
(A B) C = (A C) (B C)
C
A
B
Theorem: Let A, B, and C be any sets. Then
(A B) C = (A C) (B C)
C
A
B
Theorem: Let A, B, and C be any sets. Then
Functions
1
Acknowledgement
Most of these slides were either created by
Professor Bart Selman at Cornell University or
else are modifications of his slides
2
Functions
f(x)
Suppose we have:
x
How do you describe the yellow function?
Whats a function ?
f(x
More Counting
B
A
f
This Lecture
We will study how to define mappings to count.
There will be many examples shown.
Bijection rule
Division rule
More mapping
Counting Rule: Bijection
If f is a bijection from A to B,
then |A| = |B|
B
A
f
Power Set
How ma
Sets
1
Acknowledgement
Most of these slides were either created by
Professor Bart Selman at Cornell University or
else are modifications of his slides
2
Set Theory - Definitions and notation
A set is an unordered collection of objects referred to as eleme
CS103
Winter 2016
Handout 24
February 5, 2016
Guide to Inductive Proofs
Induction gives a new way to prove results about natural numbers and discrete structures like games,
puzzles, and graphs. All of the standard rules of proofwriting still apply to indu
Partial Orders
Motivating Example (1)
Consider the renovation of Avery Hall. In this
process several tasks were undertaken
Remove Asbestos
Replace windows
Paint walls
Refinish floors
Assign offices
Move in office furniture
Partial Orders
2
Motivating Exa
Pigeonhole Principle
f( ) =
A
B
Functions
Informally, we are given an input set,
and a function gives us an output for each possible input.
input
function
output
The important point is that there is only one output for each input.
We say a function f maps
Relations
Representing Relations
We have seen one way to graphically
represent a function/relation between two
(different) sets: Specifically as a directed graph
with arrows between nodes that are related
We will look at two alternative ways to
represen
Theorem: Let R be a binary relation over a set A. If R
is asymmetric, then R is irreflexive.
Proof: Let R be an arbitrary asymmetric binary relation
over a set A. We will prove that R is irreflexive.
To do so, we will proceed by contradiction. Suppose
tha
Divide-&-Conquer Algorithms
& Recurrence Relations:
Selected Exercises
Exercise 10
Find f( n ) when n = 2k, where f satisfies the recurrence
relation f( n ) = f( n/2 ) + 1, with f( 1 ) = 1.
Copyright Peter Cappello
2
Exercise 10 Solution
We are asked for
Discrete Mathematical Structures
Lecture 1
Slide are taken from CMU 15-251
Great Theoretical Ideas in Computer Science
https:/www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251-f09/Site/
Pancakes With A Problem!
The chefs at our place are sloppy:
when the
Predicate Logic and Quantifies
Outline
Introduction
Terminology:
Propositional functions; arguments; arity; universe of
discourse
Quantifiers
Definition; using, mixing, negating them
Logic Programming (Prolog)
Transcribing English to Logic
More ex