Discrete Mathematics I MATH/CSCI 2112 Fall 2016
Assignment 7 Due: 23 Nov 2016
(1) (a) The government of Elbonia has decided to issue currency only in 5 @ and 9 @ denominations. Show that there is largest @ value that Elbonians cannot pay with this
denomin

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 2
Due in: Wednesday 24th January, 1:30 PM
Compulsory questions
1 Which of the following are true when A = cfw_0, 1, 2 and B = cfw_1, 2, 3, 4?
Justify your answers.
(a) (n A)

Discrete Mathematics I
MATH/CSCI 2112 Assignment 1 Due 19 Sept 2014
(1) Write negations of each of the following:
(a) Roses are red and violets are blue.
(b) The bus is late or my watch is slow.
(c) If a number is prime then it is odd or it is 2.
(2) Give

Discrete Mathematics I MATH/CSCI 2112
Assignment 6 Due Fri 14 Nov 2014
Fall 2014
(1) In his book Liber Abaci (1202), Leonardo of Pisa wrote: A pair of rabbits is placed
in a walled enclosure to nd out how many ospring this pair will produce in the
course

Discrete Mathematics I MATH/CSCI 2112
Assignment 7 Due Mon 23 Nov 2015
Fall 2015
(1) (a) The government of Elbonia has decided to issue currency only in 5 @ and 9 @ denominations.
Show that there is largest @ value that Elbonians cannot pay with this deno

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 2 Due Mon, 28 Sept 2014
(1) For each statement below, give its negation and then determine if the statement or
its negation is true
(a) a, b R, ab = a b
y
(b) x Z such that y Z, x Z
(c) b R such

Discrete Mathematics I
MATH/CSCI 2112 Assignment 8 Due 8 April 2016
(1) ('3) Obtain and prove the rule for divisibility by 3 (ii) Obtain and prove the
rule for divisibility by 9
ANSWER:
Let n I (dkdk_1 ' ' ' Gilda) Z (ilk X 10k + dk_1 X 10k'1 + dk_2 X 10"

Discrete Mathematics I
MATH/CSCI 2112 Assignment 3 Solutions 1 Feb 2017
(1) The digits 0, . . . , 5 are to be used to make 4-digit numbers. Explain and find
how many such numbers can be made if:
(a) . . . repetition is allowed?
ANSWER:
The leftmost positi

Discrete Mathematics I
(
(
(
(1
MATH/CSCI 2112 Assignment 5 Solutions Due: (
Wed
Mar Fri 3 Mar 2017
(1) (a) Write the converse of: If x and y are odd, then xy is even. Is the statement
that you wrote down T or F? Prove your answer.
ANSWER:
Converse: If x

CSCI 1101
Computer Science II
PRACTICE SET FOR TEST NO.1
SOLUTIONS
1 . For each of the following questions, select the most appropriate answer.
1. A class in java is like
a. a variable
b. an object
c. an instantiation of an object
d. a blueprint for creat

Lecture Summary: 16/01/17 MATH/CSCI 2112 Winter 2017
Covered in Class:
Recap Necessary & sufficient conditions: Examples
Ex 1. If f is differentiable at a then f is continuous at a. p q.
In this case, p : f is differentiable at a is a sufficient condition

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 3 Due Wed 15 Oct 2014
n
(1) (a) Let n N. Which is larger:
answer.
r=0
2n + 1
r
2n+1
or
j=n+1
2n + 1
? Prove your
j
(n + 1)
n
= (n + 1)
.
r
(r 1)
(Hint: consider the case of selecting a committe o

Discrete Mathematics I MATH/CSCI 2112
Assignment 4 Due Wed 21 Oct 2014
Fall 2014
(1) Let n Z+ with prime factorization n = pn1 pn2 n3 . . . pnk . How many positive
2 3
1
k
divisors of n (including 1 and n) are there? Prove your answer.
(2) Given: m, k N;

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 5 Due Fri 31 Oct 2014 - no extensions
(1) Let a, b Z+ . Dene L = cfw_sa + tb | s, t, Z be the set of all linear combinations of a, b. Let L+ = cfw_x L | x > 0 (i) Verify gcd(a, b) L+ (ii) Show
z

Discrete Mathematics I
MATH/CSCI 2112 Assignment 7 Due 24 Nov 2014
(1) Show F3n is even.
(2) Prove that F1 + F3 + F5 + . . . + F2n1 = F2n
(3) Find a closed form formula for:
a0 = 2, a1 = 2 and an = 2an1 + 15an2 for n 2
(4) Dene a sequence of integers Hn b

Discrete Mathematics I MATH/CSCI 2112
Assignment 3 Due Fri 23 Oct 2015
n
(1) (a) Let n N. Which is larger:
r=0
answer.
2n + 1
r
2n+1
or
j=n+1
Fall 2015
2n + 1
? Prove your
j
ANSWER:
We expand the terms of each summation, except that we (i.e. me and my
tap

Discrete Mathematics I MATH/CSCI 2112 Fall 2015
Solutions to Assignment 5 Due 30 Oct 2015
(1) Construct 2015 consecutive composite integers (recall, the problem done in class
only had 999 consecutive integers, not 1000).
ANSWER:
To obtain 2015 consecutive

Discrete Mathematics I MATH/CSCI 2112
Assignment 6 Due 6 Nov 2015
Fall 2015
(1) (a) Prove that for positive integers, m, n; gcd(m, n) = gcd(m n, n) ANSWER:
This is a rehash of the proof of Lemma 2.
Pf: Let gcd(m, n) = d gcd(m n, n) = c.
d | m d | n d | (m

Discrete Mathematics I MATH/CSCI 2112 Winter 2016
Assignment 6 Due Fri 18 March 2016
(1) a, b, (1,? E Z 3 a = bq+ 'r'. Prove or disprove the following: cfw_1' gcd(a, q) = gcd(q, 1')
(ii) scdmm) | 0 (iii) sedan 5) = gcd(a,q) (iv) gcdm, 'r) | q
ANSWER: (2')

Discrete Mathematics I
MATH/CSCI 2112 Assignment 5 Due: Mon 7 March 2016 (hard deadline)
(1) (a)
(b)
(2) (a)
(b)
In the previous assignment, you showed: (12 even => (1 even.
Use the result to show (13 even => (1 even.
ANSWER:
cfw_13 even .e.cfw_a2 -a) eve

(1)
(2)
Discrete Mathematics I MATH/CSCI 2112 Winter 2016
Assignment 7 Due Wed 30 March 2016
In the lecture notes (29/02 - 2 f 03 Proofs: Number Theory 2), we obtained a rule for
an integer to be divisible by 11. However, the rule was obtained by the obse

Discrete Mathematics I MATH/CSCI 2112 Lecture 30/01n 01/02
Counting 3: Combinations continued and the Pigeon-hole principle
Combinations: Number of ways to choose a subset of r objects out of a set of n
objects without repetition (0 r n) is:
n
n!
n
=

Discrete Mathematics I MATH/CSCI 2112 Lecture 23-25/01
Counting Read Ch. 3 Sec 3.1 - 3.3 BoP
Count of subsets
How many subsets of does a set of n elements have?
S = cfw_s1 , s2 , . . . sn
n
0
1
2
Subsets
# subsets
cfw_
1 = 20
cfw_, cfw_s1
2 = 21
cfw_,

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 7
Due in: Wednesday 14th March, 1:30 PM
Compulsory questions
1 Solve the following recurrence relations. i.e. nd an explicit formula for
an in terms of n, and prove that it

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 9
Due: Wednesday 28th March: 1:30 PM
Compulsory questions
1 Let A = cfw_0, 1, 3, 5, B = cfw_x R|1 < x
0.3 x 3. Find:
5, and C = cfw_x R|x <
(a) A B
(b) B C
(c) A \ B
(d) A (

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 1
Due in: Wednesday 17th January, 1:30 PM
Compulsory questions
1 Rewrite these sentences symbolically:
(a) Maths is fun but Dr Kenney is not a good lecturer.
(b) If I work v

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 6
Due in: Monday 12th March, 1:30 PM
Compulsory questions
1 Show that if m > 1 and n > 1 are natural numbers such that 6|mn, then
it is possible to cover an m n chessboard w

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 10
Due: Wednesday 4th April: 1:30 PM
Compulsory questions
1 For each of the following relations, determine which of the four properties: reexivity, symmetry, antisymmetry, a

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 5
Due in: Friday 16th February, 1:30 PM
Compulsory questions
1 Use Euclids algorithm to nd the greatest common divisor of the following
pairs of numbers. Write down all the

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 3
Due in: Wednesday 31st January, 1:30 PM
Compulsory questions
1 Prove or disprove the following directly from the denitions:
(a) For any odd positive integer n, at least on

THE IMPLICATIONS OF IMPLICATIONS
MATH/CSCI2112
03 Lec 13/01
To review why an implication (p q) is true when both
the antecedent/premise (p) and the consequent/conclusion (q) are false:
Consider: If this card is a Heart then it is a Queen. Under what circu

Lecture Summary: 20/01/17 MATH/CSCI 2112 Winter 2017
Quantifiers
A proposition that has variables into which we can substitute values is called
a predicate. The truth value of the predicate depends on the values substituted
Ex. Q(n) : n2 + n + 41 is prim

Discrete Mathematics I MATH/CSCI 2112 Lecture 30/01n 01/02
Counting 3: Combinations continued and the Pigeon-hole principle
Combinations: Number of ways to choose a subset of r objects out of a set of n
objects without repetition (0 r n) is:
n
n!
n
=

MATH/CSCI 2112 Test 1 Review problems.
Here are some problems that cover the range of material covered.
The list below assumes that you have worked on the in class problems AND looked at assignment solutions.
This list is meant as a guide only.
` cfw_ in