Lecture 11
Mathematical Induction
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Oh No! Not Induction
CSCI 1900
Lecture 11 - 2
Lecture Introduction
Reading
Rosen - Section 5.1
When to consider using induction
Format for an inductive proof
Lecture 6
Integers
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Section 4.1
Remainder Theorem
Divisibility of integers
Prime numbers
GCD
LCM
Representing integers in different bases
CSCI 1900
Lecture 6
Lecture 3
Operations on Sets
CSCI 1900 Mathematics for
Computer Science
Spring 2014
Bill Pine
Lecture Introduction
Reading
Basic set operations
Rosen Section 2.2
Union, Intersection, Complement, Symmetric
Difference
Addition principle for sets
Introductio
Lecture 21
Paths and Circuits
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Section 10.1
Application of Graph Theory
Euler Paths and Circuits
Hamiltonian Paths and Circuits
Traveling Salesman Problem
CSC
Lecture 20
Finite State Machines
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Section 13.2
Machines
Finite state machines (FSM)
Examples of FSM
Model of a newspaper vending box
Model of regular expressi
Lecture 2
Introduction To Sets
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen - Section 2.1
Set Definition and Notation
Set Description and Membership
Power Set and Universal Set
Venn Diagrams
CSCI 1900
L
Lecture 4
Sequences
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Sequences
Rosen - Section 2.4
Definition
Properties
Recursion
Characteristic Function
CSCI 1900
Lecture 4 - 2
Sequence
A sequence is an ordered
CSCI 1900 - Homework 9-B
Section 1.1, 1.2, 1.4:
(22)
Let
p: I go jogging.
q: It is raining
r: Burf does not go jogging
s: I am awake
1. Write the following in terms of p, q, r and logical connectives (4)
a. Burf and I go jogging
b. I am not awake or I go
CSCI 1900 - Homework 10-B
Section 1.7, 1.8: Methods of Proof
(28)
In problems 1 through 5, give the symbolic form of the given argument, state whether the argument is valid
or not. If it is valid, either state the name of a known form or give the truth ta
CSCI 1900 - Homework 11-B
Section 5.1: Mathematical Induction
(20)
1. Given the expression F(n) = 1 + 8 + 27 + 64 + + n3
a. What is the 5th term in the expression? (1)
b. What is the kth term in the expression? (1)
c. What is the value of F(4)? (1)
d. Wri
Lecture 8
Introduction to Logic
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Logical Statements
Logical Connectives / Compound Statements
Rosen - Section 1.1
Negation
Conjunction
Disjunction
Truth Tables
Quan
Lecture 10
Methods of Proof
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen - Section 1.7, 1.8
The Nature of Proofs
Components of a Proof
Rule of Inference and Tautology
Proving equivalences
modus ponens
I
Lecture 19
Minimal Spanning Trees
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Sections 11.4, 11.5
Spanning tree
Weighted graphs
Minimal spanning tree
Two algorithms to generate
Prim's algorithm
Kruskal
Lecture 18
Tree Traversal
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Tree traversal
Rosen Section 11.3
Preorder
Inorder
Postorder
Encoding
Huffman encoding
CSCI 1900
Lecture 18 - 2
Tree Traversal
Trees can
Lecture 17
Trees
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Sections 11.1, 11.2
Review Graphs
Trees
Rooted Trees
Specialized Trees
CSCI 1900
Lecture 17 - 2
Review of Directed Graphs
A Digraph, G=(V, A
Lecture 16
Complexity of Functions
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Examine some functions occurring
frequently in Computer Science
Characterize the efficiency of an algorithm
Rosen Sections 3.2,
Lecture 15
Functions
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen - Section 2.3
Definition of a function
Representation of a function
Composition
Special types of functions
Theorems on functions
CSCI 19
Lecture 12
Permutations and Combinations
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen - Section 6.1
Learning to count sequences under four
situations:
Order matters, duplicates allowed
Order matters,
Lecture 13
Elements of Probability
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Rosen Section 6.2, 6.3
Sample spaces
Events
Probability
Pigeonhole principle
CSCI 1900
Lecture 13 - 2
Sample Space
Sample Space
Lecture 7
Matrices
CSCI 1900
Mathematics for
Computer Science
Fall 2014
Bill Pine
Lecture Introduction
Reading
Definition of a matrix
Examine basic matrix operations
Addition
Multiplication
Transpose
Bit matrix operations
Rosen - Section 2.6
Meet
Join
Mat
CSCI 1900 - Homework 8-B
Section 1.1: Propositional Logic
(22)
1. Which of the following are statements of proposition? (3)
a. The first test in CSCI 1900 is scheduled for 13 February.
b. x2 - x >12121
c. Quit talking in class.
2. Negate the following: (4
CSCI 1900 - Homework 7-B
Section 2.6: Matrices
Let
(20)
1. What is a23? (1)
2. What is a12? (1)
3. What is b31? (1)
4. What is c23? (1)
5. What is c32? (1)
6. What is c22? (1)
7. Is C a diagonal matrix? (1)
8. Given
Find the values for a, b, c, and d. (4)
CSCI 1900 - Homework 6-B
Section 4.1: Properties of the Integers (19)
1. For the integers m=33, n=9, determine the integers q and r (0 r < n) such that
m = q*n + r. (2)
2. For the integers m=117, n=13, determine the integers q and r (0 r < n) such that
m
Name _
ALS - 15
Lecture 15: Functions
/16
A function f is a relation from A to B such that each element in Dom(f) _.
Other names for function are _ and _
A function is a subset of _.
A function cannot have two ordered pairs with _
The function that maps a
Name _
ALS - 14
Lecture 14: Relations
/18
The set of all ordered pairs of elements from two sets is called the_.
A Cartesian product of m sets results in the set of _
| A1 A2 Am | = _.
A partition P of a set A is a collection of sets such that _ and
_
For
Name _
ALS - 12
Lecture 12: Permutations and Combinations
/16
The four types of sequences derived from sets can be classified with the properties _
and _ .
What values of these properties does a phone number have? _, _.
For a two-element sequence where du
Name _
ALS - 10
Lecture 10: Methods of Proof
Proof using truth tables is an example of proof by _.
In the statement (p1 p2 . pn) q, the pis are called the _ and q is called the
_.
After the statement p q is shown to be true, it is called a _.
If p q is tr