Discrete Mathematics
Introduction
Saad Mneimneh
1
Introduction
College mathematics will often focus on calculus, and while it is true that calculus is the most important eld that started modern mathematics, it is very
technical. For example, it takes a lo
CSCI 150 Discrete Mathematics
Homework 7
Due 4/21/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: An interesting language
Consider a language that uses the alphabet cfw_0, 1, #. In this language words
obey one single rule: a # cannot
CSCI 150 Discrete Mathematics
Homework 5
Due 4/5/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: Lattice points
We have five lattice points (points with integers as coordinates) in two-dimensional
space. Show that the midpoint of one
CSCI 150 Discrete Mathematics
Homework 1
Due 2/25/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: An identity
(a) Show by manipulating the expressions that:
n
n+2
= n2
2
(b) Explain in English why this is so by relating the quantitie
CSCI 150 Discrete Mathematics
Homework 3
Due 3/15/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: Prove in two ways
Prove the following:
n
k
n
=
n1
k1
k
(a) Algebraically
(b) By a combinatorial argument (count in two ways)
Problem 2:
CSCI 150 Discrete Mathematics
Homework 6
Due 4/14/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: Program termination
Consider the following program:
(w,x,y,z)=(cfw_1,.,n,cfw_1,.,n,cfw_1,.,n,cfw_1,.,n)
while w>0 and x>0 and y>0 and z
CSCI 150 Discrete Mathematics
Homework 5
Due 3/22/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1
Find the truth table for P Q in the notes.
(a) Prove that
(P Q) (P Q)
By case analysis of P
Using a truth table
(b) Prove that for any
Discrete Mathematics
Graphs
Saad Mneimneh
1
Vertices, edges, and connectivity
In this section, I will introduce the preliminary language of graphs. A graph
G = (V, E ) consists of a set of vertices V and a set of edges E , where an edge
is an unordered pa
Discrete Mathematics
Number theory
Saad Mneimneh
1
Divisibility and primes
The focus of this entire note is on positive integers. I will start by the basic
notion of divisibility. We say that a divides b, or a is a divisor of b, or b is a
multiple of a, i
Discrete Mathematics
Recurrences
Saad Mneimneh
1
What is a recurrence?
It often happens that, in studying a sequence of numbers an , a connection
between an and an1 , or between an and several of the previous ai , i < n, is
obtained. This connection is ca
Discrete Mathematics
Inductive proofs
Saad Mneimneh
1
A weird proof
Contemplate the following:
1
1+3
1+3+5
1+3+5+7
1+3+5+7+9
=
=
=
=
=
.
.
.
1
4
9
16
25
It looks like the sum of the rst n odd integers is n2 . Is it true? Certainly
we cannot draw that conc
Discrete Mathematics
Two useful principles
Saad Mneimneh
1
The inclusion-exclusion principle
I have 50 pairs of socks of which 30 are black and 35 are cotton. How many
pairs of socks are black and cotton? If I call the set of black socks A and the
set of
Discrete Mathematics
What is a proof?
Saad Mneimneh
1
The pigeonhole principle
The pigeonhole principle is a basic counting technique. It is illustrated in its
simplest form as follows: We have n + 1 pigeons and n holes. We put all the
pigeons in holes (i
Discrete Mathematics
Counting
Saad Mneimneh
1
n choose k
Consider the problem of seating n people on n chairs. In how many ways can
we do that? Lets come up with an algorithm that generates a seating. Our
algorithm consists of n stages. In stage 1 we choo
CSCI 150 Discrete Mathematics
Homework 2
Due 3/8/2016
Saad Mneimneh
Computer Science
Hunter College of CUNY
Problem 1: Round table
Consider n people to be seated on a round table with n chairs. How many
possible seating are there if:
(a) Two seatings are