University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2014
Assignment #1: Counting / Arrangements
Due: October 6, 2014 at 11:59 p.m.
This assignment is worth 10% of your nal grade.
Warning: Your electronic submissio

Assignment 4
1. Prove that the sum of the rst n odd natural numbers equals n2 .
Solution.
Let P (n) be:
The sum of the rst n odd natural numbers equals n2 , ie.
n1
i=0
2i + 1 = n2 .
Base Case n=1 then 1 = 12 .
k1
i=0
I.H. Let k N. Assume that P (k) holds,

University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2014
Assignment #3: Boolean Logic / Proof Techniques
Due: November 21, 2014 at 11:59 p.m.
This assignment is worth 10% of your nal grade.
Warning: Your electroni

University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Midterm examination
Duration:
90 minutes
Date and Time: Saturday 9 November, 5:156:45 p.m.
Aids allowed: None; closed-book, no calculators.
Make sure that y

University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2014
Assignment #4: Simple & Strong Induction / Trees
Due: December 1, 2014 at 11:59 p.m.
This assignment is worth 10% of your nal grade.
Warning: Your electroni

CSCA67 Worksheet Logical Connectives, Implication
1
Working With Propositional Statements
Exercise. Use truth tables to show that the following pairs of statements are equivalent:
(s t) and s t
(s t) and s t
Exercise. Write English sentences that illustra

ON
S
University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2014
Assignment #2: Probability / Scheduling
Due: November 4, 2014 at 11:59 p.m.
This assignment is worth 10% of your nal grade.
Warning: Your electronic su

CSCA67 Worksheet Proof by Induction
1
Proof by Induction
Lets prove the commonly used summation formula. Let S(n) refer to the
statement:
0 + 1 + 2 + + (n 1) + n =
n(n + 1)
.
2
or
n
X
i=0
i=
n(n + 1)
2
We will prove that for all n N, S(n) holds.
Proof by

2014-10-23
CSCA67 - Proofs!
Goals!
Problem Solving
!
Apply known problem solving techniques such as
greedy algorithms. !
Learn to reduce problems to ones we already can
solve.!
Formalize our language to make proofs easier to write.!
!
Proving!
!
Under

Cliques in Graphs
Denition. A clique in a graph is a set S of vertices such that every pair of vertices in S are adjacent. If the clique has n vertices,
it is denoted by Kn .
K5
Finding maximal sized cliques in large graphs is a challenging
problem in dat

Topological Orderings
Problem.
Your company needs a new website. There will be many
pages with links between them. The public is itching to view
your website so you want to publish each page as soon as
it is ready.
Caveat: You dont want any broken links.

Binary Trees
Denition A binary tree is a rooted tree in which each internal
vertex has two children and one parent.
Denition A binary search tree is a binary tree in which each
vertex contains a value and the vertices or nodes of the tree are
ordered acco

Counting Pizza Toppings*
The commercials deal was:
2 pizzas
up to 5 toppings on each
11 toppings to choose from
all for $7.98 (back in 1997).
The commercials math kid claimed there are 1,048,576 possibilities.
Lets do the calculation ourselves.
Q. How

Q. What is prob(sum=7) if we use S as the sample space?
A. 1/11
Q. Why doesnt this make sense?
A.
More than one way to roll 7
We can use S as our sample space but need to be a little more
careful.
Exercise. If we did use S = cfw_2, 3, 4, 5, 6, 7, 8, 9, 10

Sum and Product Rules
Exercise. Consider tossing a coin ve times. What is the probability of getting the same result on the rst two tosses or the last
two tosses?
Solution.
Let E be the event that the rst two tosses are the same and F
be the event that th

Counting With Repetitions
The genetic code of an organism stored in DNA molecules consist of 4 nucleotides:
Adenine, Cytosine, Guanine and Thymine.
It is possible to sequence short strings of molecules.
One way to sequence the nucleotides of a longer st

History of Pascals Triangle
Pascal was not the rst to discover the triangle of binomial coefcients but was given credit because of how he related it to his
work with probability and expectation.
The triangle may have rst appeared more than 300 years earli