The Role of Logic and Proof in Teaching Discrete Mathematics

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The Role of Logic and Proof in Teaching Discrete
Mathematics

• Summer Workshop on Discrete Mathematics Messiah
  College
• June 2006
• Susanna S. Epp

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Why Discrete Math?
The most important difference between the
demands of computer science and those of
traditional scientific or engineering
disciplines on mathematics is that, to a much
greater extent than in other disciplines,
abstraction is an essential tool of every
computer scientist, not just of the theoretician.
The computer scientist is not simply a user of
mathematical results he must use his
mathematical tools in much the same way as a
mathematician does. . . The most important
contribution a mathematics curriculum can make to
computer science is the one least likely to be
encapsulated as an individual course A deep
appreciation of the modes of thought that
characterize mathematics. W. Scherlis and M.
Shaw, Mathematics curriculum and the needs of
computer science, in The Future of College
Mathematics, A. Ralston and G. Young, eds.,
Springer-Verlag, 1983.
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The Role of Proof

• What is learned when a person becomes able to
  understand and develop basic mathematical proofs?
  
• The power of certain abstract logical principles
  (e.g., modus ponens, modus tollens, universal
  instantiatiation, generalizing from the generic
  particular, )
• How to think with symbols rather than specific,
  concrete objects
• Respect for the meanings of words, careful use of
  language
• How to deal with multiple levels of abstraction,
  to move back and forth between the abstract and
  the particular
• The necessity of being able to give a valid
  reason for the correctness of each statement in a
  chain
• How to understand and build a logically connected
  chain of statements - think in a tightly
  disciplined way

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Quote
Programming reliably - must be an activity of
an undeniably mathematical nature. . . You see,
mathematics is about thinking, and doing
mathematics is always trying to think as well as
possible. Edsger W. Dijkstra (1981) "Why
correctness must be a mathematical concern." In
The Correctness Problem in Computer Science,
Robert S. Boyer and J. Strother Moore, eds.,
Academic Press, 1981.
5
Quote
The mathematics profession as a whole has
seriously underestimated the difficulty of
teaching mathematics. Ramesh GangolliMER
WorkshopMay 31, 1991
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The Mathematical Register
Mathematicians speak and write in a special
register suited for communicating mathematical
argumentsThis register uses special words as
well as ordinary words, phrases and grammatical
constructions with special meanings .
Charles Wells The Handbook of Mathematical
Discourse www.cwru.edu/artsci/math/wells/pub/abou
thbk.html
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at least most of the time most mathematicians
would agree on the meaning of most statements
made in the mathematical register. Students
have various other interpretations of particular
constructions used in the mathematical register,
and one of their (nearly always unstated) tasks
is to learn how to extract the standard
interpretation from what is said and written. One
of the tasks of instructors is to teach them how
to do that. Charles Wells The Handbook of
Mathematical Discourse www.cwru.edu/artsci/math/w
ells/pub/abouthbk.html
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Relation between Math Meaning and Everyday
Meaning

• Example
• Promise If you eat your dinner, then you'll
  get
• dessert.
• Threat If you dont eat your dinner, then
  you wont get dessert.
• Intended interpretation
• Same for both You will get dessert if and only
  if you eat your dinner.

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Mathematical Meaning of If-then
If p then q is not logically equivalent to
If not p then not q or to
If q then p. Example Statement
If a geometric figure is a square then it has
four sides. Converse If a geometric figure has
four sides then it is a square. Inverse If a
geometric figure is not a square then it does not
have four sides.
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Relation between Math Meaning and Everyday
Meaning

• In formal mathematics the words and phrases
  if- then, and, or, not, only if,
  for all, and there exists always have
  only one meaning and there is just one set of
  conventions for their usages.
• In everyday language, these words and phrases
  sometimes have meanings and usages that are the
  same as their mathematical meanings and
  usages other times they have different
  meanings and usages.
• Example (same meaning) The Red Sox will win the
  World Series only if they win the pennant.

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• Another example (same meaning) Jacob was
  supposed to go with his father to get his hair
  cut.
• Jacob What will you give me if I get my hair
  cut?
• Jacobs mother Jacob, if you get your hair cut,
  Ill let you live.
• Jacob (wide-eyed) Does that mean that if I
  dont get my hair cut you wont
  let me live?
• Jacobs mother Of course not!

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Most mathematics classes are conducted in a
mixture of the registers of ordinary and
mathematical English, and failure to distinguish
between these two can result in incongruous
errors and breakdowns in communication. David
Pimm Speaking Mathematically Communication in
the Mathematics Classroom Routledge, 1990
(paperback)
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More Examples A Sample

• All that glisters is not gold.
• There is a time for every purpose under
  heaven.
• Actual occurrence All the students left early.
• True or false? Some students left early.
• Write down all string of three 0s and 1s in
  which all the 0s lie to the left of all the
  1s Should I include 000? 111?
• Someone says All mathematicians wear glasses.
  
• What does it mean for this to be false?

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Sometimes Things Get Tricky Even for Us
Example Ill go unless it rains. What does
this mean? a) If it doesnt rain, Ill go. b)
If it rains, I wont go. c) Ill go if, and only
if, it doesnt rain.
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However in general
The way logic and language are used in the
mathematical register is closely related to the
way they are used in high-level work in general
scientific and other academic disciplines and in
the law.
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Some Specifics Negations

• To be able to reason with a (mathematical)
  statement, a person needs to know what it means
  for the statement to be true and what it means
  for it to be false.
• Importance of being able to formulate
  negations.
• Examples
• Write a negation for 1 lt x lt 5.
• Proof by contradiction, Proof by contraposition
• Meaning of one-to-one and onto for functions
• Show that a general (universal) statement is
  false by finding a counterexample

answer 1 x 5
ouch!
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Universal Instantiation
Well Duh!

• Logical Principle If a property is true for all
  elements of a set, then it is true for any
  particular element of the set.
• But! We use this principle every time we apply a
  rule from algebra.
• Example (comes up in a standard induction
  problem)
• Simplify
• (k?2k2 2) (k 2)?2k2
• Etc. (several additional uses of the principle
  before one is finished)

(k (k 2))?2k2 2



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Generalizing from the Generic Particular

• Mathematics, as a science, commenced when
  first someone, probably a Greek, proved
  propositions about any things or about some
  things without specification of definite
  particular things.
• Alfred North Whitehead (1861-1947)

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Example

• Prove The square of any odd integer is odd.
• Proof
• (2k1)2 4k 2 4k 1 2(2k 2 2k) 1

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Whats really going on?

• Prove The square of any odd integer is odd.
• Proof
• Suppose n is any particular but arbitrarily
  chosen odd integer. Show that n 2 is odd.
• By definition of odd, n 2k1 for some integer
  k. Then
• n 2 (2k1)2 by substitution
• 4k 2 4k 1
• 2(2k 2 2k) 1
• But 2k 2 2k is an integer. So by definition of
  odd, n 2 is odd as was to be shown.

Note Both the if and the only if aspects of
the definition are used.
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Goal of My Course

• Provide foundation for math and cs courses
• Learn specific mathematical topics
• Learn how to support claims with effective
  arguments demonstrate things with mathematical
  certainty
• Improve general reasoning skills
• Dont just code, sit there!

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Discrete Mathematics I II DePaul Syllabi
Quarter Courses

• These courses are intended to provide a solid
  foundation for further study of mathematics,
  programming languages, database theory, data
  structures, and analysis of algorithms.
• An aim of both courses is to develop facility
  with the basic principles of logical reasoning
  and to learn how to apply them to formulate and
  explore the truth and falsity of a variety of
  statements in mathematics and computer science.
  Proof, disproof, and conjecture all figure
  prominently, and there is a continuing emphasis
  on written and oral communication.

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Discrete Mathematics I Syllabus
CONTENT HOURS
Logic of Compound Statements 3
Logic of Quantified Statements 3
Elementary Number Theory and Methods of Proof(divisibility division theorem rational and irrational numbers floor and ceiling mod and div direct and indirect proof division into cases) 7
Applications Algorithms(division algorithm Euclidean algorithm) 1
Sequences and Mathematical Induction 3
Definitions from set theory, functions, and relations 2
Combinatorial Reasoning (counting and probability, possibility trees and the multiplication rule combinations, the binomial theorem) 6
Review, Exams 5
TOTAL HOURS 30
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Discrete Mathematics II Syllabus
CONTENT HOURS
Set Theory (basic definitions computer science examples set properties and their proofs) 4
Functions Defined on General Sets 1
One-to-one and Onto Functions, Pigeonhole Principle, Composition 5
Recursively Defined Sequences and Applications 4
Real-valued Functions of a Real Variable and Their Graphs 0-, ?-, and ?-Notations 2
Application Efficiency of Algorithms 1
Exponential and Logarithmic Orders 2
Relations on Sets Equivalence Relations 4
Graphs and Trees and Their Applications 4
Review, Exams 3
TOTAL HOURS 30
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Summary of Student Responses in a Discrete
Mathematics Course
Problem from first test (62 students) Prove that
the difference of any odd integer minus any even
integer is odd.
Outcome Frequency Relative Frequency
100 correct 12 17
Small mistakes or linguistic infelicities 17 24
Basic idea with moderate mistakes 7 10
(2k1) 2k mistake 20 2 halves 29
Got (2r1) 2s then stymied 8 11
Proof by example 2 2 halves 4
General confusion 4 6
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Examples of Student Responses in aDiscrete
Mathematics Course
Problem from first test (19 students) Write the
following statement in the form ? __ x, if __
then __ The negative of any rational number
is rational.
Outcome Frequency Relative Frequency
100 correct 8 42
? rational s x, if x is negative then x is rational. 7 37
? numbers x, if x is a negative of a rational number then x is rational. 1 5
? real numbers x, if x is negative then x is rational. 1 5
? real numbers x, if x is irrational then -x is irrational. 1 5
? real numbers x, if the negative of x is rational then x is rational. 1 5
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Examples of Student Responses in aDiscrete
Mathematics Course for Teachers
Give-away problem from final exam (34
students) Write the following statement in the
form ? __ x, if __ then __ The negative of
any rational number is rational. BUT
Almost all students did very well on other,
harder problems where greater emphasis was
placed on providing models for correct solutions.
Outcome Frequency Relative Frequency
100 correct 18 53
? rational s x, if x is negative then x is rational. 13 38
? x ? R, -x is rational. 1 3
? rational numbers x, if (-1)x then x is rational. 1 3
? x ? X, if -x ?Q then x ?Q. 1 3
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Quote
For the human soul is hospitable, and will
entertain conflicting sentiments and
contradictory opinions with much impartiality.
George Eliot, Proem to Romola (1862-63)
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Excerpt from Article

• Very few of my students had an intuitive feel for
  the equivalence between a statement and its
  contrapositive or realized that a statement can
  be true and its converse false.
• Most students did not understand what it means
  for an if-then statement to be false, and many
  also were inconsistent about taking negations of
  and and or statements.
• Large numbers used the words "neither-nor"
  incorrectly, and hardly any interpreted the
  phrases "only if" or "necessary" and "sufficient"
  according to their definitions in logic.

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Excerpt from Article

• All aspects of the use of quantifiers were poorly
  understood, especially the negation of quantified
  statements and the interpretation of
  multiply-quantified statements.
• Students neither were able to apply universal
  statements in abstract settings to draw
  conclusions about particular elements nor did
  they know what processes must be followed to
  establish the truth of universally (or even
  existentially) quantified statements.
• Specifically, the technique of showing that
  something is true in general by showing that it
  is true in a particular but arbitrarily chosen
  instance did not come naturally to most of my
  students.
• Nor did many students understand that to show the
  existence of an object with a certain property,
  one should try to find the object.

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Examples
1. Knowing that If an integer n is
not divisible by any prime number less
than or equal to the square root of n, then n is
prime means the same as If an
integer n is not prime, then n is divisible by
some prime number less than or equal
to n. 2. Knowing that The negative
of any irrational number is irrational means
the same as No matter what irrational
number you pick, if you multiply it by
1, the result will also be irrational. And
understanding that This is false if
you can find an irrational number whose
negative is rational.
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Examples
3. Student says An integer is even if it equals
2k. My response Is 1 an even number?
Does 1 2k ?
But So its pretty important
for k to be an integer! 4. Ask student Is 0
even? Is 0 positive? Is irrational?
In fact, what is a real number? an integer? a
rational number?
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Summary

• Thesis The primary value of a discrete math
  course that is specifically addressed to freshman
  and sophomore students is that it can be
  structured so as to address students' fundamental
  misconceptions and difficulties with logical
  reasoning and improve their general analytical
  abilities.
• However It is not easy to change students'
  deeply embedded mental habits!

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My Philosophy in a Nutshell

• Teach logical reasoning, not just logic as a
  subject
• Be conscious of the tension between covering
  topics and developing students' understanding
• Don't rush to present topics from an advanced
  perspective
• Be aware that most students today are not very
  good at algebra
• In general Much miscommunication occurs
  because of unjustified assumptions

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OK - Really in a Nutshell
Interaction! Interaction! Interaction!
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Why Discrete Math?
The most important difference between the
demands of computer science and those of
traditional scientific or engineering
disciplines on mathematics is that, to a much
greater extent than in other disciplines,
abstraction is an essential tool of every
computer scientist, not just of the theoretician.
The computer scientist is not simply a user of
mathematical results he must use his
mathematical tools in much the same way as a
mathematician does. . . The most important
contribution a mathematics curriculum can make to
computer science is the one least likely to be
encapsulated as an individual course A deep
appreciation of the modes of thought that
characterize mathematics. W. Scherlis and M.
Shaw, Mathematics curriculum and the needs of
computer science, in The Future of College
Mathematics, A. Ralston and G. Young, eds.,
Springer-Verlag, 1983.
37
Uses of Web Resources a Few Examples

• As demonstrations/motivation/efficient way to
  bring certain concepts and/or situations to life
• --Tower of Hanoi--sieve of Eratosthenes--vegeta
  rians and cannibals, wolf-goat-cabbage--compariso
  n of sorting algorithms--Euclidean
  algorithm--RSA cryptography
• For extra practice
• --online exercises (e.g., tilominos, Ensley,
  Velleman)
• For self-study/tutorial/course projects/extra
  credit
• --Logic Café, slogic
• As basis for a laboratory
• --Chinese remainder theorem webpage
• --Doug Baldwins website