Lessons from Discrete Mathematics Kirsten Nelson Carleton University October 14, 2017 Contact: kirsten.nelson@carleton.ca Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 1 / 63
Introduction 1 Biography Discrete vs. Discreet, and vs. Continuous Ramsey Numbers 2 Modeling and Graph Theory Isomorphism Factorials and ‘choose’ Covering Arrays 3 Orthogonal Arrays Example and Application Programming Tools Backtracking Algorithm Data Structures Kirkman’s Schoolgirl Problem 4 Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 2 / 63
A Short Bio Cool things I’ve been able to do: Mathematician Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University ◮ Bishop Strachan School and University of Toronto Schools - CS and Math, Grades 7 to 12 Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University ◮ Bishop Strachan School and University of Toronto Schools - CS and Math, Grades 7 to 12 ◮ Independent Learning Centre, Homework Help, TVO Mathematician ◮ B. Math, C&O/PM from the University of Waterloo Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University ◮ Bishop Strachan School and University of Toronto Schools - CS and Math, Grades 7 to 12 ◮ Independent Learning Centre, Homework Help, TVO Mathematician ◮ B. Math, C&O/PM from the University of Waterloo ◮ MMT from Waterloo, 2016 Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University ◮ Bishop Strachan School and University of Toronto Schools - CS and Math, Grades 7 to 12 ◮ Independent Learning Centre, Homework Help, TVO Mathematician ◮ B. Math, C&O/PM from the University of Waterloo ◮ MMT from Waterloo, 2016 ◮ M. Sc. in Discrete Mathematics from Carleton, 2018 (hopefully) Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
A Short Bio Cool things I’ve been able to do: Programmer ◮ Netron - small software company, mostly business applications ◮ IBM - data mining software for WebSphere Commerce ◮ Intelliware - small programming-for-hire company Teacher ◮ B. Ed. from York University ◮ Bishop Strachan School and University of Toronto Schools - CS and Math, Grades 7 to 12 ◮ Independent Learning Centre, Homework Help, TVO Mathematician ◮ B. Math, C&O/PM from the University of Waterloo ◮ MMT from Waterloo, 2016 ◮ M. Sc. in Discrete Mathematics from Carleton, 2018 (hopefully) ◮ PhD??? Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 3 / 63
diagram courtesy of Math by Tori, at kelsoemath.blogspot.ca Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 4 / 63
Learning through Wikipedia? Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 5 / 63
Maybe if I just click on all the links... Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 6 / 63
A More Intuitive Description In any group of six people, there are either three people that know each other, or three people that don’t know each other. Knowing someone is symmetric; if Anthony know Betty, then Betty must know Anthony Obvious group-work opportunity “Why?” is the interesting question Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 7 / 63
Modeling with Graphs In the sense I’ll be talking about today, a graph is a series of points and lines. The points are formally known as vertices (singular vertex ), and represent any discrete set of things. The lines are formally known as edges , and represent relationships between the things. Common examples include computer networks, with computers joined by cables; cities joined by roads; and airports joined by flights. In this case we’ll be representing the people by points and the relationships by lines. If two people know each other, we’ll join them with a line. Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 8 / 63
Modeling the Problem It’s pretty natural to draw six dots to represent the six people. We can start with no relationships, meaning that we have six people who don’t know each other. We then want to add lines to get down to two people who don’t know each other, without connecting a group of three people who do all know each other. Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 9 / 63
“Without loss of generality...” We love to throw around the term “without loss of generality”, often shorted to simply “w.l.o.g.”. In this case we use it to argue that we can choose A-B as our first connection. You could choose any one of 15 lines to start with (why it’s 15 is an interesting thought exercise). No matter which line you pick, you can relabel the vertices so that A and B are the ends of the line. This is a natural way to introduce the notion of isomorphism . Whether two things are the same or not is a pretty tricky question, and it all depends on your definition. We like things to be cut and dried, so instead of saying things are the same, we say that they’re isomorphic under some operation. That makes it objective; if we agree on the operation, we (usually) agree on whether two things are isomorphic. Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 10 / 63
Back to the model We connect A and B, then consider our next move. A little thought will show you that there are two unique choices next. We can either draw our new line so that it connects with the previous line, or so that it doesn’t. If we choose the first possible letters for each, we end up with B-C and C-D as our only possible next two moves. Kirsten Nelson (Carleton University) Lessons from Discrete Mathematics October 14, 2017 11 / 63
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