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

TI
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S
University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #6: Permutations / Combinations
Due: November 11, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission

University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #6: Permutations / Combinations
Due: November 11, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission on MarkU

University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #8: Distributing Objects / Probability
Due: November 25, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission o

University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #7: Arrangement with Repetition
Due: November 18, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission on MarkU

University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Assignment #2: Counting / Probability
Due: December 1, 2016 at 11:59 p.m.
This assignment is worth 10% of your final grade.
Warning: Your electronic submission on MarkUs a

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University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #7: Arrangement with Repetition
Due: November 18, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission on

TI
ON
S
University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #8: Distributing Objects / Probability
Due: November 25, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic subm

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

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

Counting Pizza Toppings Continued.
Q. How do we know that our calculation is the correct answer?
A. One way to convince ourselves is try to nd another way to
count the same problem.
We counted all possible orders and then removed duplicates.
Exercise. Rec

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

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

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

CS 70
Spring 2017
Discrete Mathematics and Probability Theory
Course Notes
Lecture
17
Variance
We have seen in the previous note that if we toss a coin n times with bias p, then the expected number of
heads is np. What this means is that if we repeat the

ON
S
University of Toronto at Scarborough
CSCA67 / MATA67Discrete Mathematics, Fall 2016
Exercise #5: Induction / Basic Counting
Due: November 4, 2016 at 11:59 p.m.
This assignment is worth 3% of your final grade.
Warning: Your electronic submission on Ma

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

Counting Pizza Toppings Continued.
Q. How do we know that our calculation is the correct answer?
A. One way to convince ourselves is try to find another way to
count the same problem.
We counted all possible orders and then removed duplicates.
Exercise. R

Last Week
Exercises on Venn Diagrams, connectives
Predicates
Quantifiers
If and only if
Direct proofs
This Week
More Direct proofs
Order of Multiple Quantifiers
Modulus Example
Prime numbers Definition
Indirect Proofs Proof by Contradiction
Si

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #3: Logic and Proofs
Due: September 28, 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submission on MarkUs affirms

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #4: Indirect Proofs
Due: October 7, 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submission on MarkUs affirms tha

University of Toronto Scarborough
Department of Computer and Mathematical Sciences
December Examinations 2015
CSC A67H3 F / MAT A67H3 F
Duration3 hours
Aids allowed: None; closed-book, no calculators.
Make sure that your examination booklet has 13 pages (

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Exercise #1: Implication
Due: September 14, extended to September 16 2016 at 11:59 p.m.
This exercise is worth 3% of your final grade.
Warning: Your electronic submissi

University of Toronto at Scarborough
CSC A67/MAT A67 - Discrete Mathematics, Fall 2016
Assignment #1: Proofs
Due: October 26, 2016 at 11:59 p.m.
This exercise is worth 10% of your final grade.
Warning: Your electronic submission on MarkUs affirms that thi