RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH314  DISCRETE MATHEMATICS FOR ENGINEERS  MIDTERM TEST
March 2, 2007
INSTRUCTIONS
1. Duration: 2 hours item You are allowed one 8.5" 11" formula sheet (twosided). The information on the sheet must
RYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS MTH 314 Total marks: 80 NAME (Print): Instructions:
1 You are allowed an 8 2 11 formula sheet written on both sides.
Final Exam
April 25, 2007 Time allowed: 3 Hours
STUDENT #:
No other aids allowed.
.
Binary relations and directed graphs.
Reflexivity, symmetry, transitivity, equivalence relations.
liqt~ivalence
relations, equivalence classes, and partitions o f a set.
A~itisyrnmetry,partial and total orders. (Not Hasse diagrams or I'ER'f & CI'M)
Prac
Graphs and ~ r e e s *
Introduction to graphs, trees, paths and circuits
Adjacency matricies
Kruskal and Prims algorithms
Practice Exercises
Section
10.1
10.2
10.3
10.5
10.6
Exercises
1, 5,8, 10a, 12, 15, 18,24,36.
1,8,9, 12, 19,23,25,28.
2a, 3a, 4a, 6a,
23
V G U D ~ J
TP~VALIDc t c u 6 4 J 7 s ~G
p
p
Definition.
A n argument is a sequence of statements, and an argument f o m , zs a sequence of statement
jorms.
All statements ill an argument and all statement form. in an argz~mentfor, except for the
fin
Predicate Calculus
Introduction to set theory and set notation (part of section 6.1).
Quantifiers and manipulation of English statements and their equivalent symbolic form.
Generalised De Morgan laws, multiple quantifiers.
Counterexamples, nccessaly and
Graphs and ~ r e e s *
Introduction to graphs, trees, paths and circuits
Adjacency matricies
Kruskal and Prims algoritlms
Practice Exercises
Section
10.1
10.2
10.3
10.5
10.6
Exercises
1,5, 8, 10a, 12, 15, 18,24,36.
1,8,9, 12, 19,23,25,28.
2a, 3a, 4a, 6a,
Predicate Calculus
.
~
d set n 0 t a t i o n ( 4 ; ~ ~ ~ ~ ~
Quantifiers and manipulation of English st&e~nel~ts their equivalent symbolic form.
and
Generalised De Morgan laws, multiple quantifiers.
Counterexamples. necessary and sufficient conditions.
P
Graphs and ~ r e e s *
Introduction to graphs, trees, paths and circuits
Adjacency matricies
Kruskal and Prims algorithlns
Practice Exercises
Section
10.1
10.2
10.3
10.5
10.6
Exercises
1,5,8, 10a, 12, 15, 18,24,36.
1,8,9, 12, 19,23,25,28.
2a, 3a, 4a, 6a,
Predicate Calculus
681'~&:4
set notation
Quantifiers and lnanipulation of English st&ments and their equivalent symbolic form.
Generalised De Morgan laws, multiple quantifiers.
Countcrexamples, necessary and sufficient conditions.
d.
a
Practice Exercise
Binary relations and directed graphs.
Reflexivity, symmetry, transitivity, equivalence relations.
Equivalence relations, equivalence classes, and partitions of a set.
Antisymmetry, partial and total orders. (Not Hasse diagrams or PERT & CPM)
Practice Exer
Set Theory
Set inclusion and set identities. (Not including element method proofs).
Empty set, disjointness, partitions, power sets and cartesian products.
Russell's Paradox.
The Halting Problem.
Cardinality and computability.
The pigeonhole principle.
Pr
Number Theory and proofs
Number sets.
Direct and indirect proofs.
Primes and composites, divisibility, unique factorisation theorem.
div, mod, quotientremainder theorem.
gcd, Euclidean algorithm.
Practice Exercises
Section Exercises
2,7, 17,24,38,40,412,
Graphs and ~ r e e s *
Introduction to graphs, trees, paths and circuits
Adjacency n~atricies
Kruskal and Prims algoritluns
Practice Exercises
Section
10.1
10.2
10.3
10.5
10.6
Exercises
1, 5, 8, 10a, 12, 15, 18,24, 36,
1,8,9, 12, 19,23,25,28.
2a, 3a, 4a,

~
~
.
~
4)
.c
L
,
~
u
~.
~: (3). Y
u
u
K
~
~
~
~
~
~
~
~
~ ., ~.
.

~

~
~
~
k
~
& d o ,
~
.
.
~~
.
~
.
uA
~
Y ~ T
~
bgq
~
~
~
A
. OdiaZP~e
~
,Ah%
.
a  oa4eC/ ,
b
. a 4
.
i n
A uo r&.
B r ~ e C ! ? c r U e d : ? lo%.
h
.
.~
~
~
~
~
.
.
.
~
,&
Set inclusion and set identities. (Not including element method proofs).
Empty set, disjointness, partitions, power sets and cartesian products.
Russell's Paradox.
The Halting Problem.
Cardinality and computability.
The pigeonhole principle.
Practice Exer
Set Theory
Set inclusion and set identities. (Not including element niethod proofs).
Empty set, disjointness, partitions, power sets and cartesian products.
Russell's Paradox.
The Halting Problem.
Cardinality and computability.
The pigeonhole principle.
P
MTH 314 Course Outline
This outline may be updated during the term so pleasecheck regular&
Last updated: Dec 19,2010
(You may need to cl~ck ReloadRefiesh to get the page to update.)
on
The topics will not necessarily be covcred in the exact prder below.
S
RYERSON UNIVERSITY
DEPARTMENT
OF
MATHEMATICS
MTH 314
Final Exam
Total marks: 80
NAME (Print):
April 17, 2009
Time allowed: 3 Hours
STUDENT #:
Instructions:
1
You are allowed an 8 2 11 formula sheet written on both sides.
No other aids allowed. Electroni
RYERSDN UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH 314 NIIDTERM TEST Felt:rustrltr 9, 2011]
Thtal marks: at} Time allewed: 11!] Minutes.
Hates (Print): 5 STUDENT #:
Circle year instructers name: MBBBinggr
lrlstruetiens:
 sc mds are aimed Fer Markers u
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH314  DISCRETE MATHEMATICS FOR ENGINEERS  MIDTERM
TEST
March 2, 2012
INSTRUCTIONS
1. Duration: 1.5 hours item You are allowed one 8.511 formula sheet (twosided). The information
on the sheet must be handw
MTH 314
RYERSON UNIVERSITY
MIDTERM (Practice)  Winter 2015
Family Name (print):
Given Name (print):
Student ID #:
Signature:
Prof. Jonah Horowitz
Date: Feb. 23, 2015
(Time allowed: 90 minutes)
INSTRUCTIONS:
Read all instructions before starting.
Calcula
RYERSON UNIVERSITY
DEPARTMENT OF MATHEMATICS
MTH314 ~— DISCRETE MATHEMATICS FOR ENGINEERS
MIDTERM TEST
March 2013
INSTRUCTIONS
1. Duration: 1.5 hours item You allowed one 8.5” X 11" formula sheet (twosided). The information
on the sheet must be handwritt
MTH 314  Midterm (Practice) Solutions  Winter 2015
(1) (a) Let p, q, and r be three statements. Show that
(p q r) (p q) (p r) p (q r)
using standard logical equivalences (associativity, commutativity, etc).
You do not need to provide the names, but I wi