University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Assignment #3: Counting / Pascals Triangle
Due: November 19, 2013 at 4 p.m.
This assignment is worth 10% of your nal grade.
The cover sheet for this assignm
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University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Assignment #3: Counting / Pascals Triangle
Due: November 19, 2013 at 4 p.m.
This assignment is worth 10% of your nal grade.
The cover sheet for this as
University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Assignment #2: Strong & Simple Induction / BSTs
Due: October 28, 2013 at 4 p.m.
This assignment is worth 10% of your nal grade.
The cover sheet for this ass
University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Assignment #4: Counting / Probability
Due: December 2, 2013 at 4 p.m.
This assignment is worth 10% of your nal grade.
The cover sheet for this assignment su
University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2015
Assignment #2: Proof Olympics
Due: December 3, 2015 at 11:59 p.m. NO LATES ACCEPTED.
This assignment is worth 10% of your nal grade.
Warning: Your electronic submission
University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2015
Exercise #2: Counting with Repetition
Due: September 25, 2015 at 11:59 p.m.
This assignment is worth 3% of your nal grade.
Warning: Your electronic submission on MarkUs
1. The greatest common divisor of two positive integers a and b is the largest positive integer
that divides both a and b (written gcd(a, b). For example, gcd(4,6) = 2 and gcd(5,6) = 1.
(a) Prove that gcd(a, b) = gcd(a, b a).
Assume gcd(a,b)=f
f|a and f|b
1.
(a) Let a = A is working, b = B is working, and c = C is working. Write the three status
reports in terms of a, b, and c, using the symbols of formal logic.
A: BC
B:
AB
C:
AB
(b) Complete the following truth table:
a
b
c
A
B
C
T
T
T
F
T
F
T
T
F
F
T
F
T
1. Prove n(n + 1) is an even number for every non-negative integer n.
(a) without using induction
Assume n is even, n=2k
2k(2k+1)=2(k(k+1)
Assume n is odd, n=2k+1
(2k+1)(2k+1+1)=(2k+1)(2k+2)=2(2k+1)(k+1)
Thus n(n + 1) is an even number for every non-negat
1.
Prove that any line map can be 2-coloured
BCn=0 the basic line map can be colour by less than 2 colour
IHassume P(n) holds for arbitrary n0.
Proof: P(n+1) is shown as below, if we erase one line, it is certainly can be 2-coloured, and if we
switch the
University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2015
Exercise #3: Counting and Probability
Due: October 3, 2015 at 11:59 p.m.
This exercise is worth 3% of your nal grade.
Warning: Your electronic submission on MarkUs arms
Hello and welcome to Inexpensive Eats and prompt service. We provide a lot
of different services, you can dine in, take out and we do deliveries too. If you
are holding a home or office party and need party trails we are also available
for that.
We are lo
Trees
Defn. A tree is a connected graph without cycles or loops.
Trees are important data structures in computer science.
Youve already seen a rooted tree.
The Family Tree is a rooted tree. The top of the tree is the root
and the nodes follow a parent-chi
Q. What is prob(sum=7) if we use S as the sample space?
A.
Q. Why doesnt this make sense?
A.
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, 11, 12
as our sample space what
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
Boolean Logic
Another Scheduling Problem
Given a set of employees, schedule a meeting so that
everyone can attend. Assume the day is split up into time
slots and each person has a calendar which says whether
they are available for any given time slot.
Giv
Graph Theory
Originated in approx. 1736 when Leonhard Euler asked the Seven
Bridges of Knigsberg question about the river Pregel in the city
o
of Knigsberg where he lived in Prussia. The town had seven
o
bridges crossing the river. He asked:
Is it possibl
University of Toronto at Scarborough
CSCA67Discrete Mathematics for Computer Scientists, Fall 2013
Assignment #1: Simple Proofs / Scheduling
Due: October 3, 2013 at 4 p.m.
This assignment is worth 10% of your nal grade.
The cover sheet for this assignment
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
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
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
Last Week
Indirect Proofs Proof by Contradic6on
Indirect Proofs Proof by Contraposi6ve
This Week
One More Proof by Contraposi6ve Example
Proof by Exhaus6on ie, Proof by Cases
Proof by Induc6on
Proof b