8.4
Structural Induction
P. Danziger
1
Recursive Denitions
Sequences of numbers are not the only type of objects that can be dened recursively, many objects
may be dened in this way. One particularly useful application of recursion is to dene certain
cert
Discrete Mathematics 11 - MTH210 Dr. Changping Wang
10.5. Trees
A graph is said to be circuit-free if, and only ii it has no circuits. A graph is called
a tree if, and only if, it is circuitfree and connected. A trivial tree is a graph that
consists o
Discrete Mathematics II - MTH210
Dr. Changping Wang
Chapter 10. Graphs and Trees
10.1. Graphs: Definitions and Basic Concepts
Session 8
Page 1
Discrete Mathematics II - MTH210
Dr. Changping Wang
Session 8
Page 2
Discrete Mathematics II - MTH210
Dr. Changp
G
H
Is the graph H an induced subgraph of G?
No, because there is an edge ec in G, but not
in H.
Read the textbook!
There is a Homiltonian circuit,
as shown
They are isomorphic, see the one-to-one correspondence
from V(G) to V(H) below:
Note: You can list
RYERSON UNIVERSITY
Winter 2015
Discrete Mathematics II
MTH210
Midterm Test
Time: 90 minutes
Last Name (Print) _
First Name (Print) _
Signature: _
Student Number: _
Circle your section:
Section 011 (Lab: Thursday 9-10am)
Section 041 (Lab: Tuesday 5-6pm)
S
Chemical
Thermodynamics
Chapter 17
1
Outline
Spontaneity (17.1, 17.2)
Entropy (17.2, 17.3, 17.4, 17.6)
Gibbs Energy (17.5, 17.7, 17.8, 17.9)
You need to know all sections in this chapter.
There are no excluded sections.
2
Spontaneity Part 1
What is a ther
Discrete Mathematics II - MTH210
Dr. Changping Wang
Chapter 9. Counting and Probability
9.6. r-Combinations with Repetition Allowed
How many ways are there if repetition is allowed?
Session 15
Page 1
Discrete Mathematics II - MTH210
Dr. Changping Wang
Ses
Discrete Mathematics II - MTH210
Dr. Changping Wang
Chapter 4.8 The Euclidean Algorithm
Session 20
Page 1
Discrete Mathematics II - MTH210
Dr. Changping Wang
Session 20
Page 2
Discrete Mathematics II - MTH210
Dr. Changping Wang
Session 20
Page 3
Discrete
Discrete Mathematics 11 - MTH210 Dr. Changping Wang
5.2. Mathematical Induction
0. Principle of Mathematical Induction
Let P (n) be a mopmty (pimlimto) (Lloliiml for all integers
n. and let a be a fixed lllftgtfl'. Smiposo the following two
stzitioiii
Discrete Mathematics II - MTH210
Dr. Changping Wang
Chapter 9. Counting and Probability
9.6. r-Combinations with Repetition Allowed
How many ways are there if repetition is allowed?
Session 15
Page 1
Discrete Mathematics II - MTH210
Dr. Changping Wang
Ses
MATHEMATICAL INDUCTION
MIGUEL A. LERMA
(Last updated: February 8, 2003)
Mathematical Induction
This is a powerful method to prove properties of positive integers.
1. Principle of Mathematical Induction. Let P be a property of
positive integers such that:
Computation
P. Danziger
1
Finite State Automata (12.2)
Denition 1 A Finite State Automata (FSA) is a 5-tuple (Q, , , q0 , F ) where:
Q is a nite set, called the set of states. The elements of Q are called states
is a nite set, called the alphabet of the
3
Isomorphism
P. Danziger
1
Isomorphism of Graphs
Denition 1 Given two graphs G = (V, E) and G = (V , E ), we say that they are isomorphic if
there exist bijections f : V V and g : E E that preserve the endpoint relations of G and G .
i.e. if u V is an en
- 3.8
The Euclidean Algorithm
P. Danziger
1
Greatest Common Divisor (gcd)
Denition 1 Given two integers a and b, not both zero, the greatest common divisor of a and b,
denoted gcd(a, b), is the integer which satises the following two properties:
1. (gcd(a
3
Trees
P. Danziger
1
Trees
Denition 1
1. A graph is called circuit free if it contains no circuits.
2. A graph which is circuit free and connected is called a tree.
3. A graph which is circuit free is called a forest
(Note that a forest is a collection o
1.3
Formal Languages & Regular Expressions
P. Danziger
1
Cartesian Products
Denition 1 Let n Z+ , and let x1 , x2 , . . . , xn be n (not necessarily distinct) elements of some
set. The ordered n-tuple (x1 , x2 , . . . , xn ) consists of x1 , x2 , . . . ,
-4&8
Induction & Recursion
P. Danziger
1
Sequences & Series (4.1)
1.1
Sequences
A Sequence is an ordered set of numbers.
For example the sequence of even numbers, 2, 4, 6, 8, 10, . . .
We usually write ai , i = 1 . . . n to denote a sequence with n terms.
1
Denitions and Review
P. Danziger
1
Introductory Graph Theory
Informally a graph is a set of points joined by lines.
Denition 1
1. A graph is a pair (V, E), where V is a set of vertices (also called points), and E is a set of
edges (also called lines).
T
10.4
Finite Fields
P. Danziger
1
The Modular Congruence Relation (Review)
For a positive xed integer n we dene a mod n to be the remainder of a when divided by n. Note
that a mod n always yields a number less than n
C and Java use % to denote mod, i.e. a
1
Denitions and Review
P. Danziger
1
Eulerian Graphs
The Seven bridges of Knigsberg
o
Leonhard Euler (1707 - 1783) was one of the greatest and most prolic mathematicians of all
time. At the beginning and and of his career he worked in St. Petersburg in Ru
10.4
Cryptography
P. Danziger
1
Cipher Schemes
A cryptographic scheme is an example of a code. The special requirement is that the encoded
message be dicult to retrieve without some special piece of information, usually referred to as a
key. The key used
- 8.3
Fibonacci Sequence
P. Danziger
1
Fibonacci Sequence
Leonardo of Pisa (1175 - 1250), better known as Fibonacci was a man who really lives on the cusp
of the European and Arabic worlds. Fibonacci was the son of a merchant of Pisa, who served as
custom
Discrete Mathematics II - MTHZIO Dr. Changping Wang
W
Chapter 5. Sequences and Mathematical Induction
5.1. Sequences
1. Sequences, Summations and Products
A sequence is a l'(;)\ of numbers. If the row is infinitely
lmig then the sewimence is called an i