CS1231
Tutorial 2
Logic
It is recommended that you explore the available resources in the library and on the Internet to answer
the questions. This tutorial must be prepared in writing. Make two copies of your answers: one for you
and one for the tutor. B
Introduction
Cardinality
Infinity
13. To Infinity and Beyond
Cardinality
Terence Sim
Beyond Infinity
Introduction
Cardinality
Infinity
Beyond Infinity
Some infinities are bigger than
other infinities.
John Michael Green
1977
The Fault in Our Stars
Readin
Sequences
Summation & Product
Common sequences
Solving recurrences
5. Sequences & Recursion
Terence Sim
Application
Summary
Sequences
Summation & Product
Common sequences
Solving recurrences
Application
A mathematician, like a painter
or poet, is a maker
Terence Sim
11 Aug 2016
I can do it!
CS1231 Discrete Structures
^
Message of the Day
I can do it!
I can do it!
I can do it!
Dont feel like this
But like this
http:/www.jiujitsutimes.com/wp-content/uploads/Olympics_2012_Womens_75kg_Weightlifting.jpg
Think
CS1231
Tutorial 3
Logic
It is recommended that you explore the available resources in the library and on the Internet to answer
the questions. This tutorial must be prepared in writing. Make two copies of your answers: one for you
and one for the tutor. B
CS1231
Tutorial 1
Informal Introduction to Methods of Proof
The purpose of this tutorial is to learn to read and write proofs in the format prescribed. It is recommended that you explore the available resources in the library and on the Internet to answer
CS1010 Lecture #5
Repetition Statements
Round and round
PREVIOUS
LECTURE
SYNTAX
if ( condition ) cfw_
statements
Execute statements
if condition is true.
SYNTAX
The if and if-else Statements
if ( condition ) cfw_
statements_1
else cfw_
statements_2
Exe
CHAPTER 4 INDUCTION
SECTION 4.1 MATHEMATICAL INDUCTION
Mathematical induction is used to prove statements that asserts that P (n) is true for
all n Z+ where P (n) is a propositional function. It is an extremely important proof
technique.
PRINCIPLE OF MATH
CHAPTER 5: COUNTING
SECTION 5.1 BASICS
PRODUCT RULE
Suppose an operation can be broken down into a sequence of 2 steps. If the rst step can
be done in r ways and the second step can be done in s ways (regardless of how the rst
step was done), then the ent
CHAPTER 7
SECTION 7.1
DEFINITION:
A
GRAPH
G = (V, E) consists of:
V , a nonempty nite set of
E, a set of
VERTICES
AND
EDGES.
Each edge is associated with either one or two vertices called its
with one endpoint is called a LOOP.
ENDPOINTS.
An edge
Someti
CHAPTER 3 THE INTEGERS
SECTION 3.1 DIVISIBILITY
DEFINITION:
Let n, d Z with d = 0. We say that d DIVIDES n if n = dk for some k Z or equivalently,
n/d Z.
Other ways of saying include: n is DIVISIBLE by d, or n is a
FACTOR of n, or d is a DIVISOR of n.
MUL
CHAPTER 8 TREES
SECTION 8.1
DEFINITION:
A
TREE
is a connected (undirected) graph with no cycles.
REMARK
A tree cannot contain loops or multiple edges since they are cycles.
EXAMPLE
G1 , G2 are trees. G3 is not a tree as it contains a cycle. G4 is not a tr
Sequences
Summation & Product
Common sequences
Solving recurrences
Application
Summary
Quiz 1: What is the next number in the sequence?
1, 3, 5, 7, ?
Answer: 217341
Because:
18111 4
633885 2
n 90555n3 +
n 452773n + 217331
2
2
So, f (1) = 1,
f (n) =
f (2)
12. Graphs and Trees 2
Summary
Aaron Tan
31 October 4 November 2016
1
Summary
12. Graphs and Trees 2
10.5 Trees
Definitions: circuit-free, tree, trivial tree, forest
Characterizing trees: terminal vertex (leaf), internal vertex
10.6 Rooted Trees
Defini
Statistics for Mechanical Engineers
Problems
1. If Pr(A) = 0.1, Pr(B) = 0.2, Pr(C) = 0.3, find Pr(A B C)
i. if A, B and C are mutually exclusive;
ii. if A, B and C are independent.
2. If Pr(A) = 0.43, Pr(B) = 0.37, Pr(A B) = 0.21, find Pr(A0 ), Pr(A B 0 )
Computational Models - Lecture 2
Non-Deterministic Finite Automata (NFA)
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University.
p.1
Computational Models - Lecture 2
Non-Deterministic Finite Automata (NFA)
Closure of
Step 1:
Installation Guide
Connect your modem.
Wireless-N Gigabit Router with USB
Step 2:
Step 3:
Connect your router.
Connect your computer.
WNR3500L
NETGEA R
Internet
(not included)
DSL or Cable
CAUTION:
MAKE SURE YOUR MODEM IS
TURNED OFF OR UNPLUGGED
B
IJCEM International Journal of Computational Engineering & Management, Vol. 12, April 2011
ISSN (Online): 2230-7893
www.IJCEM.org
49
Online Polling System using UML Methodology
Lalit K Bansal1, Harshit 2
1
Department of Information Technology,
Doaba Insti
Unit 3
Algebra and Number Sense:
Proportions
and Percents
Develop an understanding
of and apply proportionality.
CHAPTER 6
Ratios and Proportions
Use ratio and
proportionality to solve problems,
including those with tables and
graphs.
CHAPTER 7
Applying P
ISSN: 2312-7694
Indrajeet et al. / International Journal of Computer and Communication System Engineering (IJCCSE)
Feasibility Study on e-Voting System
Indrajeet Sharma
Dr. Sanjay Kumar Dubey
Dept. of Computer Science and Engineering
ASET, Amity Universit
ELECTRONIC VOTING SYSTEM
Prepared By
Md. Mostafizur Rahman
STUDENT ID: 02201006
Md. Sharfuddin Bhuiyan
STUDENT ID: 02101059
Md. Rajibul Hossain
STUDENT ID: 02201010
A thesis submitted in partial fulfillment of the requirements for the
degree of Bachelor o
Assignment #5
Econ 302: Intermediate Macroeconomics
December 2, 2009
1
Keynesian Cross
Consider a closed economy.
Consumption function:
C = C + M P C(Y T )
(1)
In addition, suppose that planned investment expenditure is I , the level of taxes is T , and g
3201 Computer Networks 2014/2015
Handout: Subnetting Question
Subnetting Questions with Answers
Question1:
Given the following:
Network address: 192.168.10.0
Subnet mask: 255.255.255.224
1. How many subnets?
Ans: 6
2. How many hosts?
Ans: 30
3. What are t
A Brief Introduction to Fitting Models and Parameter Estimation
Least Squares Fitting
Suppose we have N pairs of observations, (x1 , y1 ), . . . (xN , yN ) and we expect x and y to be
related according to some function, g, which itself involves one or mor
Class Quiz
Q1
Determine whether the following data (table 1.1) support a proportionality argument for
being proportional with P0.5.
3.5
5
6
7
8
P
3
6
9
12
15
[10]
Q2
The questions on Lotka-Volterra predator-prey dynamics use the following equations.
dV/d
Internet voting feasibility study
A summary
Table of contents
Introduction . 2
System functionality. 3
System requirements . 5
Information security . 6
Additional requirements concerning information security . 6
Residual threats. . 7
Cost and benefit anal
081231
NATIONAL UNIVERSITY OF SINGAPORE
081231 - DISCRETE STRUCTURES
(SEMESTER 2 AY 2014/2015)
Time allowed: 2 hours
INSTRUCTIONS TO CANDIDATES
1. This assessment paper contains FIVE questions and cornprises EIGHT printed pages,
including this page.
Answe
Wireless Connection Guide
See over for wired instructions
Connecting to the Internet 2016/17
There are two ways to connect to the Internet in your room: the first is using eduroam
wireless and the second is by connecting to the ResNet socket on your wall.
Computational modelling techniques
Exercise set 2
Due: 15.9.2014
1. Determine whether the following data support a proportionality argument for y being
proportional with z:
Y
Z
3.5
3
5
6
6
9
7
12
8
15
2. Derive the equations that minimize the sum of the s
Congestion Control and QoS
CS158a
Chris Pollett
Apr 11, 2007.
Outline
Congestion Control Algorithms
Quality of Service
Congestion Prevention Policies
We briefly point out some policy issues in
each layer which can affect congestion.
The choice is like
Cardinality
Read Rosen, 2.1, 2.4; Epp (2nd ed.), 7.6
1
Finite and Infinite Sets
Definitions.
A set S is called finite if S is the empty set or there is a 1-to-1
correspondence from S to cfw_1, 2, . . . , n where n is a positive
integer. In the first case,
CS1231
Tutorial 7
Graph Theory
This tutorial must be prepared in writing. Make two copies of your answers: one for you and
one for the tutor. Bring your lecture notes to the tutorial. The answers to some of these questions
require the use of the theoretic
Combinations
Read Rosen, 5.3, 5.5
1
r-Combinations
Definition. An r-combination of a set of n elements is an unordered selection (subset) of r elements taken from the set of
n elements. The number of r-combinations (subsets of size r)
of a set of n elemen
CS1231
Tutorial 8
Trees
This tutorial must be prepared in writing. Make two copies of your answers: one for you and
one for the tutor. Bring your lecture notes to the tutorial. The answers to some of these questions
require the use of the theoretical resu
Chapter 7
Relations, Equivalence Relations and Orders
A friend to all is a friend to none.
Corpus Aristotelicum, by Aristotle
7.1
Relations
Ordered pairs give us the opportunity to associate mathematical objects. Sets of ordered
pairs are collections of s