Homework 4
Due date: 02/19/2017
Problem 1 (9 points)
Give an example or prove that there are none.
i)
A simple graph with 6 vertices, whose degrees are 2, 2, 2, 3, 4, 4.
Solution: None. It is not possible to have one vertex of odd degree
ii)
A simple grap
Discrete Mathematics Lecture 9
Alexander Bukharovich New York University
Graphs
Graph consists of two sets: set V of vertices and set E of edges. Terminology: endpoints of the edge, loop edges, parallel edges, adjacent vertices, isolated vertex, subgraph
A Relation on a set A is called an Equivalence Relation if it is Reexive, Symmetric and
Transitive.
At this point make sure these definitions are clear with all of them.
Two elements a and b which are related by an equivalence relation are said to be equi
Quiz 5
Total: 25 Time: 20 Minutes
Problem 1 (2+4 points)
State 2 applications of a planar graph. Determine (give reason) if this graph is a planar.
Electronic ckt.
Checking/reducing dimensentionality
No its falling k3,3
Problem 2 (1.5+1.5+3)
State the tra
Midterm Exam
March ,08 2017
nginerring
Discrete Mathematics
Total:
Time: 1 hour and 20 minutes
Name:
ID No: .
Question 1:
Solution:
Question 2
Use the Principle of Mathematical Induction to prove that 1 2 + 22 23 + + (1)n2n =
(2n+1(1)n + 1)/3 for all po
This is the submission by
Sahil Shah (netid :[email protected] N Number : N12706992)
FCS Homework 2
1.
Use the definition of big-O to prove that
Ans. Let us assume x > 1 .
To prove that the given function is big-O of
For x > k .
Since
This leads to
Where cor
This is the submission by
Sahil Shah (netid :[email protected] N Number : N12706992)
Dhruv Amin (netid: [email protected] N Number: N11713617 )
Poojan Shah (netid : [email protected] N Number : N10482975)
FCS Homework 1
Problem #1
The following proposition uses th
This is the submission by
Sahil Shah (netid :[email protected] N Number : N12706992)
FCS Homework 3
1. Determine whether the following graphs are isomorphic.
Soln:
Here the number of vertices and number of edges are same which are 6 and 10 respectively so
th
Propositional Logic
# Is the following sentence a proposition? If it is a proposition, determine
whether it is true or false.
Portland is the capital of Maine.
# Is the following sentence a proposition? If it is a proposition, determine
whether it is true
Relations
Chapter Summary
Relations and Their Properties
n-ary Relations and Their Applications
Representing Relations
Closures of Relations (not currently included in
overheads)
Equivalence Relations
Partial Orderings
Relations and Their Properties
Bi
Show All Solutions
Rosen, Discrete Mathematics and Its Applications, 6th edition
Extra Examples
Section 9.3Representing Graphs and Graph Isomorphism
Page references correspond to locations of Extra Examples icons in the textbook.
p.616, icon at Example 9
Discrete Mathematics Lecture 8
Alexander Bukharovich New York University
Recursive Sequences
A recurrence relation for a sequence a0, a1, a2, is a formula that relates each term ak to certain collection of its predecessors. Each recurrence sequence needs
Discrete Mathematics Lecture 7
Alexander Bukharovich New York University
Generic Functions
A function f: X Y is a relationship between elements of X to elements of Y, when each element from X is related to a unique element from Y X is called domain of f,
Homework 3
Problem 1 (1+5 points)
Write the algorithm for finding FIBONACCI number using recursive method. Show/prove that the
algorithm is correct.
Basis 0,1
Inductive:
(Hint: inductive fn= fn-1 + fn-2 k>=2 )
Problem 2 (5+3)
Show the binary tree and sort
Home Work 1
Total: 60
Due date: 09/14/2016
Problem #1
6 points
Write these system specifications in symbols using the propositions
v: The user enters a valid password,
a: Access is granted to the user,
c: The user has contacted the network administrator,
Home Work 1 (part2)
Total 25 points;
Problem 1 (4 points)
Show the looping steps in order to find the number 15 with binary search in the list 7 8 10 12 12
15 16 18 20. Need to show the variables in the algorithm/pseudocode and update in each step.
proced
Homework 11
Total: 45 points Deadline: 04/23/2017
Problem #1 (2 points)
Describe the difference between Prims algorithm and Kruskals algorithm for
finding a spanning tree of minimum cost.
Solution:
Using Prims algorithm, at each stage the edges selected w
Discrete Mathematics Lecture 1 Logic of Compound Statements
Alexander Bukharovich New York University
Administration
Class Web Site
http:/cs.nyu.edu/courses/summer03/G22.2340-001/index.htm
Mailing List
Subscribe at http:/cs.nyu.edu/mailman/listinfo/g22_
Discrete Mathematics
Lecture 2
Logic of Quantified Statements
Alexander Bukharovich
New York University
Predicates
A predicate is a sentence that contains a
finite number of variables and becomes a
statement when specific values are
substituted for the v
Discrete Mathematics
Lecture 3
Elementary Number Theory and
Methods of Proof
Alexander Bukharovich
New York University
Proof and Counterexample
Discovery and proof
Even and odd numbers
number n from Z is called even if k Z, n = 2k
number n from Z is c
Discrete Mathematics
Lecture 4: Sequences
and Mathematical Induction
Alexander Bukharovich
New York University
Sequences
Sequence is a set of (usually infinite number of)
ordered elements: a1, a2, , an,
Each individual element ak is called a term, wher
Discrete Mathematics Lecture 5
Alexander Bukharovich New York University
Basics of Set Theory
Set and element are undefined notions in the set theory and are taken for granted Set notation: cfw_1, 2, 3, cfw_1, 2, cfw_3, cfw_1, 2, 3, cfw_1, 2, 3, , , cfw_
Discrete Mathematics Lecture 6
Alexander Bukharovich New York University
Counting and Probability
Coin tossing Random process Sample space is the set of all possible outcomes of a random process. An event is a subset of a sample space Probability of an e
Show All Solutions
Rosen, Discrete Mathematics and Its Applications, 6th edition
Extra Examples
Section 9.5Euler and Hamilton Paths
Page references correspond to locations of Extra Examples icons in the textbook.
p.634, icon at Example 2
#1. Determine wh