1/4
CIS1910 Discrete Structures in Computing (I)
Winter 2013, Solutions to Assignment 2
PART A
1. These two binary operations are used to define the set C of complex numbers.
Let (a,b), (c,d) and (e,f
Connectivity
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Connectivity
1/7
Connectivity
Definition
Let G = (V, E) be a simple graph. A path is a sequence of vertices v0 , v1 ,
Sets and Set Operations
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Sets and Set Operations
1 / 20
Sets
Definition: Set
A set is an unordered collection of objects.
Sets are
Planar Graphs
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Planar Graphs
1/8
Planar Graphs
Circuit Design
When designing electronic circuits or computer chips, it is often des
Induction
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Induction
1 / 16
Induction
What is Induction?
I reasoning from the particular to the general; whats true for particulars
Integer Representations
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Integer Representations
1 / 12
Other Integer Representations
It is often useful to represent integers in b
Trees and Their Applications
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Trees and Their Applications
1 / 14
Trees
We already know that graphs can be used to model many probl
Introduction to Graphs
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Introduction to Graphs
1 / 10
Problems Solved Using Graphs
Graphs are frequently used to model real world p
Functions
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Functions
1 / 16
Functions
The theory that has had the greatest development in recent times
is without any doubt the the
Sequences and Sums
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Sequences and Sums
1 / 17
Sequences
Definition
Sequences are used to represent ordered lists of elements.
They
Methods of Proof
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Methods of Proof
1 / 26
Definitions
Definition
A theorem is a statement that can be shown to be true.
We demonstr
Discrete Probability
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Discrete Probability
1 / 23
Where do we use Probability?
I
Average case analysis of running times for algorit
Boolean Algebra
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Boolean Algebra
1 / 28
Outline
Topics:
I
Boolean variables, functions, operators
I
Relationship with logic
I
Combi
Introduction to Logic
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Introduction to Logic
1 / 46
Logic: What is it Good For?
The rules of logic apply directly to future topics
The Basics of Counting
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
The Basics of Counting
1 / 36
Combinatorics
Definition
Combinatorics is the study of the arrangement of obj
Hi there! Since James (so far) isnt
expanding a whole lot on the slides, Ill
just send you the slides with my
annotations for now. Let me know if
that doesnt work for you for any
reason. Ive removed a
www.menti.com : 60 12 2
Discrete Math
Quantified Statements
Quantifiers
Quantifiers provide us with a notation that allows to quantify (count)
how many objects in the domain of the variable satisfy a
www.menti.com : 60 12 2
Discrete Math
Predicates
Course Updates
A1 Posted
Rosen Course Textbook ( Unavailable )
Questions from last time
Implication operator as a normal arrow ( =>, , > ) can use any
Generalized Permutations and Combinations
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Generalized Permutations and Combinations
1 / 21
Permutations with Repetition
So far wev
Relations
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Relations
1 / 14
Relations
Common relationships:
I a business and its clients
I staff and their time sheets
I clients an
Terminology and Special Types of Graphs
CIS 2130 (DE)
Notes by: Joe Sawada
University of Guelph
CIS 2130 (DE)
Terminology and Special Types of Graphs
1 / 16
Terminology for Undirected Graphs
Definitio
1/6
CIS1910 Discrete Structures in Computing (I)
Winter 2013, Solutions to Assignment 4
QUESTION A.
1. (a) The domain of definition of f is the set of all the elements x
of the domain R such that (x1)
1/5
CIS1910 Discrete Structures in Computing (I)
Winter 2013, Solutions to Midterm
To simplify the task of marking, Q1 was marked out of 18, Q2 out of 7.5, Q3 out of 10.5,
Q4 out of 13.5 (plus 3 bonus
1/3
CIS1910 Discrete Structures in Computing (I)
Winter 2013, Solutions to Assignment 1
PART A
1. Possible answers are (a) S=cfw_1,2 (b) S=cfw_2,3 (c) S=cfw_2,cfw_3 (d) S=cfw_3,cfw_3.
For example, the
1/7
CIS1910 Discrete Structures in Computing (I)
Winter 2013, Solutions to Assignment 3
A. Rules of Inference
1) Let r be the proposition it rains, let f be the proposition it is foggy, let s be the p
Composition of Relations

The composition of relations R and S on set A is another relation on A, denoted S o R
The pair (a,c) S o R if and only if there is a,b,c A such that (a,b) R and (b,c) S
Here
QUANTIFIERS

Quantifiers provide notation that allows to quantify (count) objects in the domain of the variable
UNIVERSAL QUANTIFIERS

(upside down capital A) is known as the universal quantifier
x
SEQUENCES


A sequence is a special type of function in which the domain is a consecutive set of integers
When a function is specified as a sequence, using subscripts to denote the input to the func

Discrete mathematics is the study of mathematical structures that are fundamentally discrete
rather than continuous
LOGIC


Logic is the study of formal reasoning important to make precise stateme