Discrete Mathematics I
MATH/CSCI 2112 Assignment 2 Due 25 Sept 2015
(1) Prove that the following argument is valid (use the statement symbols shown,
in brackets, at the end). If Jose took the jewellery or Mrs. Krasov lied, then a
crime was committed. Mr.
Discrete Mathematics I
Assignment 2 / Solutions / / MATH/CSCI 2112 / / Due: 25 Sept 2015
(1) Prove that the following argument is valid (use the statement symbols shown,
in brackets, at the end). If Jose took the jewellery or Mrs. Krasov lied, then a
crim
Discrete Mathematics I
MATH/CSCI 2112 Assignment 3 Due: Friday 12 Feb 2016
(1) Consider the set cfw_1, 2, . . . , 10, 11. How many numbers do you need to pick, in the
worst case, to get a pair that adds up to 11?
ANSWER:
No number can pair with 11 so it i
Discrete Mathematics I
MATH/CSCI 2112 Assignment 2 Due 30 Jan 2015
(1) Let G represent the set of all T.V. game shows. Let P be the set of people in
your neighbourhood. Let C(p, g) be the predicate that the person p appeared
on game show g and D(p) be the
Discrete Mathematics I
MATH/CSCI 2112 Assignment 7 Due 24 Nov 2014
(1) Show F3n is even.
ANSWER:
For n = 1 , F3 = 2 hence even
Assume for k > 2, F3k is even.
F3(k+1) = F3k+3 = F3k+2 + F3k+1 = (F3k + F3k+1 ) + F3k+1 = F3k + 2 F3k+1
The rst term is even by
Discrete Mathematics I
MATH/CSCI 2112 Assignment 1 Solutions 21 Sept 2016
(1) Write negations of each of the following (hint: first write each symbolically):
(a) Roses are red, violets are blue.
ANSWER: (Using de Morgans) Roses are not red or violets are
Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 6 Solutions Due Fri 14 Nov 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 t
CSci/Math2112
(9)
Assignment 2
Due May 29, 2015
1. Negate the following statements:
(a) If for the integers x, y, z we know that x divides y and y divides z, then x divides z.
Solution: There exist integers x, y, and z such that x divides y and y divides
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.
(3)
Roses are not red or violets are not blue.
(b) The bus is late or my watch is slow.
(1)
The bus i
Discrete Mathematics I
MATH/CSCI 2112 Assignment 1 Due 20 Jan 2016
(1) (a) Given the implication p q, show that its converse and inverse are logically
equivalent.
ANSWER:
Converse: q p
Inverse: p q
Recall that p q is equivalent to the boolean expression p
Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Solutions to 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+
Discrete Mathematics I
MATH/CSCI 2112 Assignment 3 Solution Due 9 Oct 2015
Please give an explanation with each answer for full points
(1) A chess club has 2n members. They need to pair up the members for chess
matches.(i) In how many ways can they pair u
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
ANSWER: False - this only holds up to a
Discrete Mathematics I MATH/CSCI 2112
Lecture 24-26/09 Counting 2
One more PHP problem:
Ex. In a class of n students, show that there must be at least two students
with the same number of friends in the class (friendship is a reexive relationship).
So f
Discrete Mathematics I MATH/CSCI 2112
Lecture 19/09 Counting
Count of subsets
How many subsets of does a of n elements have?
S = cfw_s1 , s2 , . . . sn
n
0
1
2
Subsets
# subsets
cfw_
1 = 20
cfw_, cfw_s1
2 = 21
cfw_, cfw_s1 , cfw_s1 , s2 4 = 22
.
.
Th
Discrete Mathematics I MATH/CSCI 2112 Lecture 29/09 Counting 3
Combinations: The number of ways to choose a subset of r objects out of a set
of n objects without repetition (0 r n) is:
n
r
=
n!
r! (n r)!
(N ote : If 0 n r then
n
r
= 0)
The combination sym
Discrete Mathematics I MATH/CSCI 2112 Lecture 6, 8 Oct Proofs
and Number theory 1
Direct Proofs:
Ex 1 Try your hand at solving this: There is a black box with 60 B balls and
a white box with 60 W balls. You take 20 W balls and mix tem into the black
box.
Discrete Mathematics I MATH/CSCI 2112 Lecture 29/09 Counting 3
Combinations: The number of ways to choose a subset of r objects out of a set
of n objects without repetition (0 r n) is:
n
r
=
n!
r! (n r)!
(N ote : If 0 n r then
n
r
= 0)
The combination sym
Discrete Mathematics I MATH/CSCI 2112 09 Lec 10 Oct Proofs/NT 2
Defn: Q indicates the set of all Rational numbers.
Defn: x Q p, q Z, (q = 0)
x = p/q
Defn: A ratio p/q is in its lowest form i there is NO integer d such that xd/yq = p/q.
Ex. Prove that the
Discrete Mathematics I MATH/CSCI 2112 Lec 17 Oct NT/Proofs 3
Finding gcd(a, b): The above denitions of gcd(m, n) (not both 0), are not
ecient for computing the gcd, but rst a detour:
Thm The Division Algorithm:
For a, b N q, r Z; q, r unique
b = aq + r;
Discrete Mathematics I MATH/CSCI 2112 Lec 22 Oct Induction 1
Inductive thinking: 100 dumb (i.e. mute) logicians meet in a forest clearning at each toll of the bell. They each wear either a B hat or a G hat. They
can see everyone elses hat but not their o
Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 4 Solutions Due Wed 21 Oct 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.
ANSWER:
O
Lecture Summary: 17/01/14 MATH/CSCI 2112 Fall 2014
Quantiers
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 prime. N
Lecture Summary: 10/09/14 MATH/CSCI 2112 Fall 2014
Covered in Class:
Recap Necessary conditions, sucient conditions: Examples
Ex1. If f is dierentiable at a then f is continuous at a. p q.
In this case, p : f is dierentiable at a is a sucient condition to
IMPLICATIONS OF IMPLICATIONS
p
0
0
1
1
Given p q has the truth table:
q pq
0
1
1
1
0
0
1
1
We write the BF for p q using DNF (Disjunctive Normal Form):
p q = ( p q) ( p q) (p q)
Notice that p is common in the rst two brackets, so using the distributive ru
Discrete Mathematics I MATH/CSCI 2112 Lec 12/11 Mathematical
Induction 4
Q. How large do the Fibonacci numbers get? Can we obtain a closed form
solution?
The answer to the second question is yes. On the way to showing that, we
will answer the rst questio
MATH/CSCI 2112
Discrete Mathematics I
Fall 2014
Notes 24 Nov Solving Linear Congruences
We have seen that congruences can be added, multiplied and subtracted.
I had put o division. The time has come to deal with division of congruences.
Q. How do we cance
MATH/CSCI 2112
Discrete Mathematics I
Notes 17 Nov Congruences
Fall 2014
Recall the denition: A Realtion R from set A to set B is a subset of AB (See lec.
notes 8 Sept). Further, a function is a relation such that if x, y R x, z y = z
A relation R A A is
Discrete Mathematics I MATH/CSCI 2112 Fall 2017
Assignment 6 (Primal Solutions Due: 17 March, in class.
(1) In the proof of the infinitude of primes, we come across the integer
M = p1 p2 pk + 1, where p1 < p2 < < pk . As show in lecture (Lec 17
Feb/09Proo
Discrete Mathematics I MATH/CSCI 2112 Winter 2017
Assignment 7 Solutions Due Wed 27 March 2017
(1) Consider the proposition P (n) = n2 + 5n + 1 even00
(Note: Please answer the parts in order, else the point of the question is lost.
(a) Prove that P (k) P
Discrete Mathematics I MATH/CSCI 2112 Winter 2017
Assignment 8 Solutions Due Friday, 7 April 2017
(1) (a) The government of Elbonia has issued currency only in 5@ and 9@ denominations. Show that there is a largest @ value (nx ) that Elbonians cannot pay
w
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
Discrete Mathematics I
(
(
(
(1
MATH/CSCI 2112 Assignment 5 (Updated) 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.
(b) Prove that for a Z,