MTH 110 - Discrete Math I
Assignment 2
Due at the beginning of the lab, week of September 23
1. Write the contrapositive, converse and inverse of the following statement.
If I drink too much coffee, then I do not sleep at night.
2. (Section 2.2, Question

MTH 110 - Discrete Math I
Assignment 4
Due at the beginning of the lab, week of October 7
1. (Section 1.2, Question 7b) Use set-roster notation to indicate the elements in the set
T = cfw_m Z | m = 1 + (1)i for some integer i.
2. (Section 1.2, Question 9c

Errata
P Q (P Q) and
P|Q (P Q) so:
P Q P Q P|Q
1 1
0
0
1 0
0
1
0 1
0
1
0 0
1
1
Recall:
Let an be a sequence of real numbers and L be a real number.
Then lim an = L iff for every > 0 there is an N > 0 such that
n
whenever n > N (n Z) we have that L < an <

Definition:
A real number is rational if it can be expressed as the quotient
of two integers with a nonzero denominator. A real number that
is not rational is called irrational.
Definition:
A real number is rational if it can be expressed as the quotient

Universal Conditional Statements
Definition:
A universal conditional statement is a statement of the form
x U(P(x) Q(x).
A universal conditional statement x U(P(x) Q(x) is
equivalent to x cfw_y U|P(y)Q(x).
Notice that this is what we were doing in the pre

Tilomino
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Tilomino
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Small(k), Large(d), Medium(b), Smaller (b, a), Larger (j, k),
etc. . .
Tilomino
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x, y (SameSize(x, y) Black(y)
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Tilomino
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x, y (Same

MTH 110 - Discrete Math I
Assignment 6
Due in lab, week of November 11
1. (Section 4.6, Question 6) Formulate the negation of the following statement, and then
prove the statement by contradiction.
There is no least positive rational number.
2. (Section 4

Suppose we have a universal set U.
Commutative Laws: A B = B A and A B = B A.
Associative Laws: (A B) C = A (B C) and
(A B) C = A (B C).
Distributive Laws: A (B C) = (A B) (A C) and
A (B C) = (A B) (A C).
Identity Laws: A = A and A U = A.
Complement Laws:

Order of Operations
1)
2)
3)
4)
Brackets
Negation (~)
And, Or ()
Implies, iff (, )
So p q p q is the same as: (p q) (p q)
Summary of Logical Equivalences
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
Commutative Law
Associative Law
Distributive Law
Identity Law
Negation

Functions
Definition:
A function f from X to Y (denoted F : X Y ) is a relation
from X to Y such that every element in X is related to an
element in Y , and each element in X is not related to more
than one element in Y .
Functions
Definition:
A function

Definition:
The truth set of a predicate P is the set of elements of the
domain which make P true.
Definition:
The truth set of a predicate P is the set of elements of the
domain which make P true.
The statement x D, P(x) is true if and only if D is a sub

What is a Set?
When considering one particular set (lets name that set S),
every object is either an element of S or not an element of S.
When considering one particular set (lets name that set S),
every object is either an element of S or not an element

MTH 110 - Discrete Math I
Lab 1 - Not to be handed in.
Week of September 9
1. (Section 2.1, Question 3) Fill in the blanks so that the argument in part (b) has the
same logical form as the argument in part (a).
(a) This number is even or this number is od

Boolean Algebras
Definition:
A Boolean algebra is a set B together with two operations
(generally denoted + and ) such that for all x, y B, both
x + y, x y B and the following properties hold:
I
Commutative Laws: For every x, y B, x + y = y + x and
x y =

We can express the rational numbers as equivalence classes of
pairs of integers:
We can express the rational numbers as equivalence classes of
pairs of integers:
Let R be the following relation on Z (Z cfw_0):
(a, b)E(c, d) k Z cfw_0 such that (a, b) = (k

MTH 110 - Discrete Math I
Assignment 3
Due at the beginning of the lab, week of September 30
1. (a) Give the output signals for the circuit pictured below with input signals P = 0,
Q = 0, R = 0.
P
OR
Q
AND
AND
S
NOT
R
(b) Write the input/output table for

Answers to MTH110 midterm, Fall 2011
1.
(p q) (p q)
2. (a)
(b)
sr
s
p s
s
p
( p q) (p q)
( p q) ( p q)
p ( q q)
pc
p
(4)
(Specialization)
(1)
(a)
(Elimination)
p (q s) (2)
p
(b)
(c)
q s
(Modus ponens)
q s
s
(d)
q
(e)
sr
r
q
r
(f)
qr
(c)
(a)
(Elim

Back to Arguments
Back to Arguments
a.
b.
c.
d.
pq r
qs
(s q) (q p)
r
Back to Arguments
a.
b.
c.
d.
pq r
qs
(s q) (q p)
r
(s q) (q p)
sq
1.
(c)
(Specialization)
Back to Arguments
a.
b.
c.
d.
pq r
qs
(s q) (q p)
r
(s q) (q p)
sq
(c)
(Specialization)
(s q)

Answers to MTH110 midterm, Fall 2011
1. Draw a truth table: (we show the columns for the two statements to
be compared, but you should fill in the details)
p q r (p (q r) ( q p) (p r)
T T T
F
F
T T F
T
T
T
T
T F T
T F F
T
T
F
F
F T T
F T F
F
F
F F T
F
F
F

RYERSON UNIVERSITY
FALL 2011
Discrete Mathematics
MTH 110
Midterm Test
60 Minutes
NAME (PRINTED)
Student Number
SIGNATURE
Circle your section:
Sec. 011 (Lab: Thursday 1pm)
Sec. 021 (Lab: Wednesday 1pm)
Sec. 031 (Lab: Tuesday 11am)
Question
1
2
3
4
5
Total

RYERSON UNIVERSITY
FALL 2012
Discrete Mathematics I
MTH 110
Midterm Test
75 Minutes
NAME (PRINTED)
Student Number
SIGNATURE
Circle your section:
Sec. 011 (Lab: Friday 12pm)
Sec. 041 (Lab: Wednesday 9am)
Sec. 021 (Lab: Tuesday 12pm) Sec. 051 (Lab: Wednesda

Recall: f : X Y is one-to-one (aka injective) iff for every
x, y X , if f (x) = f (y ) then x = y .
Recall: f : X Y is onto (aka surjective) iff for every y Y
there is an x X with f (x) = y .
Example: Define f : Z+ Z+ by
f (x) = 7x
Is f onto?
Example: Def

MTH 110 - Discrete Math I
Assignment 8
Due in lab, week of November 18
1. (Section 8.1, Question 4) Define a relation R on Z as follows: For all integers m and
n,
m R n m and n have a common prime factor.
(a) Is 15 R 25?
(b) Is 22 R 27?
(c) Is 0 R 5?
(d)

MTH 110 - Discrete Math I
Worksheet Not to be handed in
To be discussed in the lab, week of October 21
1. (Section 3.2, Question 3bd, 4bd) Write a negation of each of the following statements:
(a) computers c, c has a CPU.
(b) a band b such that b has won

De Morgans Laws
(p
q) p q
(p
q) p q
De Morgans Laws
p
T
T
F
F
q
T
F
T
F
p
(p
q) p q
(p
q) p q
q
pq
p
q
(p
q)
De Morgans Laws
p
T
T
F
F
q
T
F
T
F
(p
q) p q
(p
q) p q
p
q
F
F
T
T
F
T
F
T
pq
T
F
F
F
p
q
(p
q)
De Morgans Laws
p
T
T
F
F
q
T
F
T
F
(p

The midterms will most likely be returned next Wednesday
in class.
Sets (Again)
Sets (Again)
Definition:
Let A and B be sets. Say that A and B are disjoint, if A B = .
Sets (Again)
Definition:
Let A and B be sets. Say that A and B are disjoint, if A B = .

Oops!
Definition:
An argument is sound if it is valid and all its premises are true.
Oops!
Definition:
An argument is sound if it is valid and all its premises are true.
The following argument is not sound, because it is invalid:
Oops!
Definition:
An argu

Proof Techniques in the Context of Elementary Number
Theory
What You Already Know
What You Already Know
I
The basic laws of algebra.
What You Already Know
I
I
The basic laws of algebra.
Three properties of equality:
I
I
I
For all objects A, A = A.
For all

Recall: A relation from A to B is a set of ordered pairs
R A B.
Recall: A relation from A to B is a set of ordered pairs
R A B.
Example: Let A = cfw_2, 3, 4, 7, 13, B = cfw_27, 36, 65, 77 and say
that
xRy p Z such that p is prime, p < x and p|y
Draw the a

MTH 110 (DISCRETE MATHEMATICS I) COURSE NOTES
9.
27
November 8, 2016
Introduction to Sets
Definition 9.1. A set is a collection of objects called elements. The cardinality |S| of a
set S is the number of elements in the set (could be finite or infinite).

4
PAWEL PRALAT
2.
September 13, 2016
Example 2.1. There are some other important logical equivalences:
Double negative law: ( p) p
De Morgans laws: (p _ q) p^ q, (p ^ q) p_ q
Commutativity: p _ q q _ p, p ^ q q ^ p
Associativity: p _ (q _ r) (p _ q) _

MTH 110 (DISCRETE MATHEMATICS I) COURSE NOTES
4.
13
September 27, 2016
Computer Representation of Negative Integers
Definition 4.1. The twos complement of a positive integer a (relative to a fixed bit length
n) is the n-bit binary representation of 2n a.

30
PAWEL PRALAT
10.
November 15, 2016
Example 10.1. Show that S = (A \ B) \ (A \ B) = ;.
First proof: For a contradiction, suppose that x 2 S. Then x 2 (A \ B) and x 2 (A \ B).
But this implies that x 2
/ B and x 2 B. Contradiction.
Second proof:
(A \ B)

MTH 110 (DISCRETE MATHEMATICS I) COURSE NOTES
3.
9
September 20, 2016
Example 3.1. Show that the following argument is valid:
(1) p_ s
(2) p ! (q_ s)
(3) u ! ( q_ r)
(4) s ^ r
) q^ u
by using standard argument forms (Modus Ponens, Modus Tollens, etc.) and

24
PAWEL PRALAT
8.
November 1, 2016
Modular Arithmetic
Example 8.1. 7 kids are sharing 4 bags of candy. After they divided up the candy, there is
one piece left over. How many pieces of candy were in each bag?
Possible answers: 2, 9, 16, 23, . . . . One c