UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F - Discrete Mathematics Final Examination December 7, 2005 SOLUTIONS
Duration: 2.5 hours
This is a "closed book" examination; no aids are permitted. No electronic or mechanica
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ECE190F Discrete Mathematics
Midterm Test
October 24th, 2005
Examiners: John Carter, Paul McCabe
SOLUTIONS
This is a closed book test; no aids are permitted.
No electronic or mechanical co
ECE 190 - Fall, 2006 Assignment 4 Solutions
Section 3.1
26 Suppose that m is any odd integer and n is any even integer. By definition of odd and even there exist integers p and q such that m = 2p+1 and n = 2q. Then m-n = 2p+1-2q = 2(p-q)+1. Since p
ECE 190 - Fall, 2006 Assignment 3 Solutions
Section 1.3
36 See text. 42 from (b) and (d) and modus tollens, we can conclude q - (1) From (1) and (a) and elimination, we can conclude p - (2) From (e) and (1) and modus ponens, we can conclude u s -
ECE 190 - Fall, 2006 Assignment 2 Solutions
Section 1.1
6 (a) s i (b) s i 8 (a) h w s (b) w h s (c) h w s (d) w s h (e) w (h s) 15 p T T F F q T F T F p q T F F F (p q) F T T T p q T T T F (p q) (p q) T T T T
32 Thi
ECE 190 - Fall, 2006 Assignment 1 Solutions
3 only A and D are equal B and C have R for their universe and -1 B but -1 C / E = D because 0 E (its universe is Z+ , not Z) / 4 (a) S = {-1, 1} (b) T = {0, 2} (c) U = - no integer is both 2 and -2
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F - Discrete Mathematics Final Examination December 18, 2006 SOLUTIONS
Duration: 2.5 hours
This is a "closed book" examination; no aids are permitted. No electronic or mechanic
33
33.1
Planar Graphs
Introduction to planar graphs
definition a graph is planar if it can be drawn in the plane without any intersecting edges. concept of a face in a planar graph a face is a region of the plane bounded by the edges and vertices
30
30.1
Hamiltonian Circuits and Weighted Graphs
Hamiltonian circuits
definition: a circuit that contains every vertex exactly once (but not necessarily every edge) examples: using K5 and a subset of K5
. . . . . . . . . . . . . . . . . . . . . .
27
27.1
Introduction to Graphs
Graph basics
a graph is a set of the form G = (V, E) V is a set of vertices (or nodes) E is a set of edges representation: draw a picture v1
v4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
24.1
Relations
Introduction
consider sets A = {Mon, Ott, Tor, Van}, B = {BC, ON, QC} recall: the Cartesian product A B = {(a, b)|a A, b B} = {(Mon, BC), (Mon, ON), . . ., (Van, ON), (Van,QC)}. suppose we are only interested in the ordered pa
ECE 190 - Fall, 2006 Assignment 6 Solutions
Section 6.2
20 (a) first digit cannot be zero 9 10 10 10 = 9000 (b) first digit cannot be zero and last digit must be odd 9 10 10 5 = 4500 (c) first digit cannot be zero, other digits cannot repeat 9
ECE 190 - Fall, 2006 Assignment 7 Solutions
Section 6.3
5(a) To be divisible by five, the last digit must be 0 or 5. The first digit cannot be zero so there are 9 10 10 10 2 = 18 000 such integers. 11 See text. 21 (a) See text. 21 (c) From pa
ECE 190 - Fall, 2006 Assignment 8 Solutions
Section 6.9
1 See text. 2 P (X Y ) P (Y ) P (X Y ) = P (X|Y )P (Y ) 1 1 = 3 4 1 = 12 P (X|Y ) = 4 See text. 5 See text. 7 (a) See text. Since both of the people chosen are treated equally (they both
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ECE19OS — Discrete Mathematics
Final Examination
‘ May 1, 2006
Examiner: John Carter
Duration: 2.5 hours
o This is a “closed book” examination; no aids are permitted.
o No electronic or mech
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ECE190H1 — Discrete Mathematics
Final Examination
April 27, 2007
Examiner: Mahdi Shabany
Duration: 2.5 hours
0 This is a “closed book" test; no aids are permitted.
0 No electronic or mechani
"Finitary" mathematics versus Mathematics of "infinity"
Periklis A. Papakonstantinou
University of Toronto
This is an informal discussion regarding the distinction of Discrete Mathematics from other branches of Mathematics. Most Discrete Math texts
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F - Discrete Mathematics Midterm Test October 25th, 2006 Examiners: John Carter, Periklis Papakonstantinou SOLUTIONS
This is a "closed book" test; no aids are permitted. No ele
ECE 190 - Fall, 2004 Assignment 12 Solutions
Questions for Week of November 27 to December 1 Section 11.5
8 See text. 9 See text. 10 See text. 11 See text. 12 See text. 15 One such graph is
. . . . . ..
. . . . . . .
20 One such graph is
.
ECE 190 - Fall, 2006 Assignment 10 Solutions
1. a) Only K2 is bipartite. K1 is not considered to be bipartite - there is only one vertex so we cannot place the vertices in two non-empty sets. For n 3, Kn will always contain a triangle so it cannot b
22
22.1
Probability (cont.)
Independent events: P (A B) = P (A)P (B) - Epp 6.9
events A and B are independent iff P (A B) = P (A)P (B) an example consider 3-child families assign equal probabilities to GGG, GGB, . , BBB (all 1 ) 8 let A be: 1st
16
16.1
Counting (cont.)
Solving problems by applying basic principles
often there are restrictions e.g. third letter must be A or E first digit must not be zero usually best to deal with restrictions first often helpful to use boxes e.g. arrange
UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING ECE190F - Discrete Mathematics Midterm Test October 28th, 2004 Examiners: John Carter, Ben Liang
Duration: 1 hour, 50 minutes (7:10pm to 9:00pm)
This is a "closed book" test; no aids
A little-bit of Graph Theory
Periklis A. Papakonstantinou
University of Toronto
We consider the problem of extracting global properties for a graph by just knowing some local information about it. This will take us to the first simpler but yet remar