Discrete Structures

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Discrete Structures AlgorithmsPropositional
Logic

• EECE 320 UBC Spring 2009

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Summary so far

• Pancake sorting
• A problem with many applications
• Bracketing (bounding a function)
• Proving bounds for pancake sorting
• You can make money solving such problems (Bill
  Gates!)
• Illustrated many concepts that we will learn in
  this course
• Proofs
• Sets
• Counting
• Performance of algorithms

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Propositional logic

• Logic of compound statements built from simpler
  statements using Boolean connectives.
• Building block for mathematics and computing.
• Direct applications
• Design of digital circuits
• Expressing conditions in programs
• Database queries

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A sandwich is better than God

• Nothing is better than God.
• A sandwich is better than nothing.
• Thus, a sandwich is better than God.

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What is propositional logic?

• The simplest form of mathematical logic.
• Develops a symbolic language to treat compound
  and complex propositions and their logical
  relationship in an abstract manner.
• And, before we get ahead of ourselves... what is
  a proposition?
• A declarative statement that is either true or
  false (but not both!).

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Examples of propositions

• September 6 2007 is a Thursday.
• September 6 2007 is a Friday.
• 32 equals 7.
• There is no gravity.

• The following are not propositions.
• Do your homework.
• What is the time?
• 34.

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Propositions and their negations
Suppose p is a proposition, then the negation of
p is written as p.
The negation of proposition p implies that It is
not the case that p. Examples p It is raining.
p It is not raining. p 325. p 32?5.
Notice that p is a proposition too.
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Conjunctions and disjunctions

• Elaborate ways of saying and and or
• Consider two propositions, p and q
• Conjunction (and) p ? q
• It is a bright and windy day.
• The day has to be both bright and windy.
• Disjunction (or) p V q
• To ride the bus you must have a ticket or hold a
  pass.
• One of the two conditions (have a ticket or
  hold a pass) suffices. (Though both could be
  true.)

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Exclusive or

• In day-to-day speech, sometimes we use or as an
  exclusive or.
• I will take a taxi or a bus from the airport.
• Only one of taxi or bus is implied.
• To be precise, one would need to say I will take
  either a taxi or a bus from the airport.
• Exclusive or XOR is denoted by the symbol.

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Truth tables

• Truth tables can be used to evaluate statements.
  A simple proposition can either be true or
  false.

Notice that the conjunction and disjunction of
two propositions are also propositions (and,
along with negation, are called compound
propositions).
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Logical equivalence
Two statements are logically equivalent if and
only if they have identical truth tables. The
simplest example is (p) p.
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Implication

• Another important logical construct is
  implication
• This is akin to saying If ... then ...
• When proposition p holds then q holds.
• Notation p ? q.
• Example If I am on campus, I study.
• p I am on campus.
• q I study.

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Truth table for implication

• p ? q If p, then q.
• If p is true, q must be true for the implication
  to hold.
• p is the assumption/premise/antecedent.
• q is the conclusion/consequent.

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An equivalent for implication
Is there an expression that is equivalent to p ?
q but uses only the operators , ?, V?
Consider the proposition p V q
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Proving other equivalences

• Easy to use truth tables and show logical
  equivalence.
• Example distributivity
• p ? (q V r) (p V q) ? (p V r)
• Do this as an exercise.
• You would have seen these forms in earlier
  courses on digital logic design.

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Logical equivalences

• The basic laws
• Identity
• Domination
• Idempotence
• Negation and double negation
• Commutation
• Association
• Distribution
• Absorption
• De Morgans laws

Propositions that are logically equivalent. You
will need to know them, although we will not
elaborate on them in lecture. In the text
(Rosen) Chapter 1, Section 2.
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Variations on a proposition

• Given a proposition p ? q, there are other
  propositions that can be stated.
• Example If a function is not continuous, it is
  not differentiable.
• Contrapositive q ? p
• Example If a function is differentiable, then it
  is continuous.
• Converse q ? p
• Example If a function is not differentiable,
  then it is not continuous.
• Inverse p ? q
• Example If a function is continuous, then it is
  differentiable.

Of the three (contrapositive, converse, inverse),
which is not like the other two?
Hint One is a logical equivalent of the original
proposition.
The contrapositive is equivalent to the
original proposition.
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Tautology Contradiction

• A proposition that is always true is called a
  tautology.
• Example p V p
• A proposition that is always false is called a
  contradiction.
• Example p ? p

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Bi-conditionals (if and only if)

• If p and q are two propositions, then p ? q is a
  bi-conditional proposition.
• p if and only if q. (p iff q)
• p is necessary and sufficient for q.
• If p then q, and conversely.
• Example The Thunderbirds win if and only if it
  is raining.
• p ? q is the same as (p ? q) ? (q ? p).
• Are there other other equivalent statements?

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What is an argument?

• A sequence of statements the ends with a
  conclusion.
• (Not the common language usage of a debate or
  dispute.)

Structure of an argument Statement 1
(p1) Statement 2 (p2) ... Statement n
(pn) Statement n1 (conclusion)
premises or antecedents
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Example

• Premises
• If you have a current password, you can log onto
  the computer network.
• You have a current password.
• Conclusion
• Therefore, you can log onto the computer
  network.

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Representing an argument
An argument is sometimes written as follows
premises
conclusion
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Valid arguments

• An argument is valid if and only if it is
  impossible for all the premises to be true and
  the conclusion to be false.
• How do we show that an argument is valid?
• We can use a truth table, or
• We can show that (p1 ? p2 ? ... ? pn ? pn1) is a
  tautology using some rules of inference.

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Why use rules of inference?

• Constructing a truth table is time consuming!
• If we have n propositions, what is the size of
  the truth table? 2n, which means that the table
  doubles in size with every proposition.

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Rules of inference
modus ponens method of affirming
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Rules of inference
modus tollens method of denying
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Rules of inference
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Rules of Inference
More rules of inference listed in the text. Try
proving them as an exercise. Chapter 1, Section 5
(Rosen).

• p?q Rule of hypothetical
  q?r syllogism?p?r
• p ? q Rule of disjunctive ?p syllogism? q

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Inference Rules for Quantifiers
More rules of inference listed in the text. Try
proving them as an exercise. Chapter 1, Section 5
(Rosen).

• ?x P(x)?P(o) (substitute any specific object o)
• P(g) (for g a general element.)??x P(x)
• ?x P(x)?P(c) (substitute a new constant c)
• P(o) (substitute any extant object o) ??x P(x)

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Formal Proofs

• A formal proof of a conclusion C, given premises
  p1, p2,,pn consists of a sequence of steps, each
  of which applies some inference rule to premises
  or previously-proven statements to yield a new
  true statement (the conclusion).
• A proof demonstrates that if the premises are
  true, then the conclusion is true.

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Formal Proof Example

• Suppose we have the following premisesIt is
  not sunny and it is cold.We will swim only if
  it is sunny.If we do not swim, then we will
  canoe.If we canoe, then we will be home
  early.
• Given these premises, prove the theoremWe will
  be home early using inference rules.

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Proof Example cont.

• Let us adopt the following abbreviations
• sunny It is sunny cold It is cold swim
  We will swim canoe We will canoe early
  We will be home early.
• Then, the premises can be written as(1) ?sunny
  ? cold (2) swim ? sunny(3) ?swim ? canoe (4)
  canoe ? early

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Proof Example cont.

• Step Proved by1. ?sunny ? cold Premise 1.2.
  ?sunny Simplification of 1.3. swim?sunny Premise
  2.4. ?swim Modus tollens on 2,3.5. ?swim?canoe
  Premise 3.6. canoe Modus ponens on 4,5.7.
  canoe?early Premise 4.8. early Modus ponens on
  6,7.

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Proofs and theorems
A theorem is a statement that can be shown to be
true. A proof is the means of doing so.
Axioms, postulates, hypotheses and previously
proven theorems
Rules of inference
Proof
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Common Fallacies

• A fallacy is an inference rule or other proof
  method that is not logically valid.
• A fallacy may yield a false conclusion!
• Fallacy of affirming the conclusion
• p?q is true, and q is true, so p must be true.
  (No, because F?T is true.)
• Example
• If you do every problem in the book, then you
  will learn mathematics. You learned mathematics.
• Therefore you did every problem in the book
  (incorrect!).

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Common Fallacies

• Fallacy of denying the hypothesis
• p?q is true, and p is false, so q must be
  false. (No, again because F?T is true.)
• Example
• If you do every problem in the book, then you
  will learn mathematics. You did not do every
  problem in the book.
• Therefore you did not learn mathematics
  (incorrect!).

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Common fallacies Circular Reasoning

• The fallacy of (explicitly or implicitly)
  assuming the very statement you are trying to
  prove in the course of its proof. Example
• Prove that an integer n is even, if n2 is even.
• Attempted proof Assume n2 is even. Then n22k
  for some integer k. Dividing both sides by n
  gives n (2k)/n 2(k/n). So there is an integer
  j (namely k/n) such that n2j. Therefore n is
  even.
• Circular reasoning is used in this proof. Where?

Begs the question How doyou show that jk/nn/2
is an integer, without first assuming that n is
even?
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A Correct Proof

• Prove that an integer n is even, if n2 is even.

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Proof Methods for Implications

• For proving implications p?q, we have
• Direct proof Assume p is true, and prove q.
• Indirect proof Assume ?q, and prove ?p.
• Vacuous proof Prove ?p by itself.
• Trivial proof Prove q by itself.
• Proof by cases Show p?(a ? b), and (a?q) and
  (b?q).

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Direct Proof Example

• Definition An integer n is called odd iff n2k1
  for some integer k n is even iff n2k for some
  k.
• Theorem (For all numbers n) If n is an odd
  integer, then n2 is an odd integer.
• Proof If n is odd, then n 2k1 for some
  integer k. Thus, n2 (2k1)2 4k2 4k 1
  2(2k2 2k) 1. Therefore n2 is of the form 2j
  1 (with j the integer 2k2 2k), thus n2 is
  odd. ?

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Indirect Proof Example

• Theorem (For all integers n) If 3n2 is odd,
  then n is odd.
• Proof Suppose that the conclusion is false,
  i.e., that n is even. Then n2k for some integer
  k. Then 3n2 3(2k)2 6k2 2(3k1). Thus
  3n2 is even, because it equals 2j for integer j
  3k1. So 3n2 is not odd. We have shown that
  (n is odd)?(3n2 is odd), thus its
  contra-positive (3n2 is odd) ? (n is odd) is
  also true. ?

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Vacuous Proof Example

• Theorem (For all n) If n is both odd and even,
  then n2 n n.
• Proof The statement n is both odd and even is
  necessarily false, since no number can be both
  odd and even. So, the theorem is vacuously true.
  ?

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Trivial Proof Example

• Theorem (For integers n) If n is the sum of two
  prime numbers, then either n is odd or n is even.
• Proof Any integer n is either odd or even. So
  the conclusion of the implication is true
  regardless of the truth of the antecedent. Thus
  the implication is true trivially. ?

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Proof by Contradiction

• A method for proving p.
• Assume ?p, and prove both q and ?q for some
  proposition q. (Can be anything!)
• Why does it work?
• Thus ?p? (q ? ?q)
• (q ? ?q) is a trivial contradiction, equal to F
• Thus ?p?F, which is only true if ?pF
• Thus p is true.

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Proof by Contradiction Example

• Theorem sqrt(2) is irrational.
• Proof
• Assume 21/2 were rational. This means there are
  integers i,j with no common divisors such that
  21/2 i /j.
• Squaring both sides, 2 i2/j2, so 2j2 i2. So
  i2 is even thus i is even.
• Let i 2k. So 2j2 (2k)2 4k2. Dividing both
  sides by 2, j2 2k2.
• Thus j2 is even, so j is even.
• But then i and j have a common divisor, namely 2,
  so we have a contradiction. ?

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Review Proof Methods So Far

• Direct, indirect, vacuous, and trivial proofs of
  statements of the form p?q.
• Proof by contradiction of any statements.
• Next Constructive and nonconstructive existence
  proofs.

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Proving Existence

• A proof of a statement of the form ?x P(x) is
  called an existence proof.
• If the proof demonstrates how to actually find or
  construct a specific element a such that P(a) is
  true, then it is a constructive proof.
• Otherwise, it is nonconstructive.

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Constructive Existence Proof

• Theorem There exists a positive integer n that
  is the sum of two perfect cubes in two different
  ways
• equal to j3 k3 and l3 m3 where j, k, l, m are
  positive integers, and j,k ? l,m
• Proof Consider n 1729, j 9, k 10, l
  1, m 12. Now just check that the equalities
  hold.

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Nonconstructive Existence Proof

• Theorem There are infinitely many prime
  numbers.
• Any finite set of numbers must contain a maximal
  element, so we can prove the theorem if we can
  just show that there is no largest prime number.
• I.e., show that for any prime number, there is a
  larger number that is also prime.
• More generally For any number, ? a larger prime.
• Formally Show ?n ?pgtn p is prime.

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The proof, using proof by cases...

• Given ngt0, prove there is a prime pgtn.
• Consider x n!1. Since xgt1, we know (x is
  prime)?(x is composite).
• Case 1 x is prime. Obviously xgtn, so let px
  and were done.
• Case 2 x has a prime factor p. But if p?n, then
  p mod x 1. So pgtn, and were done.

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Pancake numbers

• How did we prove the bounds on Pn?
• n ? Pn ? 2n 3
• What are the propositions involved?

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Wrap up

• Propositional logic
• Or propositional calculus
• Truth tables
• Logical equivalence
• Basic laws
• Rules of inference
• Arguments
• Premises and conclusions
• Proofs