Lectures 11 and 12
We discussed the cannonball stacking problem.
We showed how it was reasonable to suspect that the number of cannonballs
n(n + 1)(2n + 1)
in a square based stack of height n is given by
.
6
We proved this statement by Mathematical Induct
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for Homework 2
1. Exercise 24 on page 126.
Solution:
a) The power set of every set includes at least the empty set, so the
power set cannot be empty. Thus is not the power set of any
set.
b) This
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Homework 3, due Monday, September 21
READING: Chapter 1, 2, 3, 4.
1. Exercise 8 on page 216. (15 points)
2. Describe an algorithm for nding the two smallest integers in a nite
sequence of distinct integers
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
(Last) Homework 5, due Monday, October 12
READING: Chapters 6, 7, 9 and 10.
1. Exercise 28 on page 451. (15 points)
2. In an experiment you pick at random a bit string of length 5. Consider
the following e
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for Homework 4
1. Exercise 14 on page 330.
Solution:
Basis Step: For n = 1 we get 1 21 = (1 1)21+1 + 2, which is true
statement 2 = 2.
Inductive Step: Let us assume the inductive hypothesis, name
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for Homework 5
1. Exercise 28 on page 451.
Solution:
First solution: We need to nd the number of ways for the computer to select its 11 numbers, and we need to nd the number
of ways for it to se
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Practice Final Exam
These problems are sample problems for the nal exam, so you may expect
similar problems in the nal. Do not hand in your solutions. Solutions will
be handed out, discussed (and posted on
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for the Midterm Exam
1. Are (p q) r and p (q r) logically equivalent? Justify your
response.
Solution: No, for example if they are all false, then the rst one is
false but the second one is true.
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Practice Midterm Exam
These problems are sample problems for the midterm exam, so you may
expect similar problems in the midterm. Do not hand in your solutions.
Solutions will be handed out and discussed (
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for the Practice Final Exam
These problems are sample problems for the nal exam, so you may expect
similar problems in the nal. Do not hand in your solutions. Solutions will
be handed out, discus
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for Homework 1
1. Exercise 14 on page 13.
Solution:
(a) r q
(b) p q r
(c) r p
(d) p q r
(e) (p q) r
(f) r (q p)
(20 points)
2. Construct truth tables for each of the following compound propositio
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Homework 1, due Tuesday, September 8
Most homeworks will be worth 100 points; consider the point value in
determining how much time you spend on each question. The exercise and
page numbers are from the 7t
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for Homework 3
READING: Chapter 1, 2, 3, 4.
1. Exercise 8 on page 216.
Solution: In each case we have to nd the least integer n such that
f (x) is O(xn ) so there must be constants C and k such t
In lectures 3 and 4 we discussed the meaning of sets, and proper notation
for sets. We dened the sets of natural numbers N, integers Z, rational numbers
Q, and the set of real numbers R,
We discussed what it meant for a set to be well-dened.
We discussed
Exercises from Lectures 1 and 2
Discreteness
1. Children at a birthday party take turns striking a piata lled with candy
n
with a stick. What aspects of this situation are discrete? What aspects
are continuous. To what extent could discrete mathematics be
In lectures 5 and 6 we discussed the operations on sets: union, intersection,
complement and cartesian product.
We also discussed the cardinality of sets, set identities, the the double inclusion method of showing two sets are equal.
Lastly we discussed t
Lectures 7 and 8
In lectures 7 and 8 discussed countability.
We showed by Cantors Diagonal Argument that the subsets of a countable
set are uncountable.
So |N| < |P(N)|.
We showed how to compare cardinalities of innite sets with functions:
If there exists
Lectures 15 and 16 Introduction to number theory.
We introduced division with remainder.
We showed that to check whether a number n is prime, we need only check
its divisibility with respect to all primes up to n.
We introduced and explained the Sieve of
Lectures 9 and 10
We discussed formal logic.
We introduced the idea of Boolean variables, which model statements, and
Boolean functions, which model logical expressions.
We compared real values, constants, and functions, with Boolean values,
constants and
Lectures 13 and 14 Binomial Coecients and
Pascals Triangle
We used induction to prove that the binomial coecients satisfy:
|Pk (A)| =
|A|
k
=
n!
.
k!(n k)!
The induction proof gave us an inductive formula:
n
n
+
k1
k
=
n+1
k
which led to arranging the coe
Lectures 17 and 18 Number Theory II
Introduced the notation a | b for a divides b.
Dened to notions:
p is irreducible:
For all a and b, p = ab (a = 1) (b = 1)
and p is prime:
For all x and y, p | xy (p | x) (p | y)
Proved easily that p is irreducible: p i
Lectures 19 and 20 Number Theory III
We examined exponents and discovered that while
(a a mod p) (b c mod p) (a + b a + b mod p)
and
(a a mod p) (b c mod p) (ab a b mod p)
for exponents
(a a mod p) (b c mod p) (ab (a )b mod p)
It turns out the exponents i
CS 2022/ MA 2201 Discrete Mathematics
A term 2015
Solutions for the Practice Midterm Exam
These problems are sample problems for the midterm exam, so you may
expect similar problems in the midterm. Do not hand in your solutions.
Solutions will be handed o