Math 2183

1
Math 2183

• Discrete Structures

2
Sec 1.1

• Propositional Logic

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Definitions Propositions and Compound
Propositions

• A proposition p is a statement or declarative
  sentence that is either true or false, but not
  both.
• ?p The negations of p denotes the proposition
  It is not the case that p.
• p?q The proposition p and q denotes the
  proposition that is true when both p and q are
  true and false otherwise.
• p?q The proposition p or q denotes the
  proposition that is true only when both p and q
  are true and false otherwise.
• ?

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Compound Propositions

• p ? q The proposition p exclusive or q
  denotes the proposition that is true when exactly
  one of p and q is true and false otherwise.
• p ? q The implication p ? q denotes the
  proposition that is false when p is true and q is
  false and true otherwise. p is called the
  hypothesis or antecedent or premise q is called
  the conclusion or consequence.
• p ? q The biconditional p ? q denotes the
  proposition (p ? q) ? (q ? p). It is true when p
  and q have the same truth values, and false
  otherwise.
• ?

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Truth Tables
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Equivalent Statements

• Definition Two propositions are equivalent if
  they have the same truth values.
• An Implication p ? q
• Its Inverse ?p ? ?q
• Its Converse q ? p
• Its Contrapositive ?q ? ?p
• Theorem The implication and its contrapositive
  are equivalent.
• Theorem The inverse and the converse of any
  implication are equivalent.

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Homework

• Sec 1.1
• pg 16  1, 3, 5, 7(a,b,c), 9(b,c,d), 11(a,c,e)
  15, 19, 21(a,b), 23(a,b), 27(a,b,c), 29(a,c,e)

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Sec 1.2

• Logical Equivalence

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Definitions

• A compound proposition that is always true is
  called a tautology.
• A compound proposition that is always false is
  called a contradiction.
• A compound proposition that is neither a
  tautology or a contradiction is called a
  contingency.
• The propositions p and q are called logically
  equivalent (written p ? q ) if p?q is a tautology
• ?

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Logical Equivalences

• Identity Laws p ? T ? p p ? F ? p
• Dominance Laws p ? T ? T p ? F ? F
• Idempotent Laws p ? p ? p p ? p ? p
• Double Negative Law ?(? p) ? p
• Commutative Laws p ? q ? q ? p p ? q ? q ? p
• Associative Laws (p ? q) ? r ? p ? (q ? r)
• (p ? q) ? q ? p ? (q ? r)
• Distributive Laws p ? (q ? r) ? (p ? q) ? (p ?
  r)
• p ? (q ? r) ? (p ? q) ? (p ? r)
• ?

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Logical Expressions

• De Morgans Laws ?(p ? q) ? ?p ? ?q
• ?(p ? q) ? ?p ? ?q
• Absorption Laws p ? (p ? q) ? p
• p ? (p ? q) ? p
• Negation Laws p ? ?p ? T
• p ? ?p ? F
• Equivalence p ? q ? ?p ? q
• p ? q ? ?q ? ?p
• ?

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Homework

• Sec 1.2
• pg 28 1(c,e), 3a, 4a, 5, 7(b,d), 9(b,d), 15,
  17, 21, 29

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Sec 1.3

• Predicates and Quantifiers

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Introduction

• For statements such as John is a computer
  science major John is called the subject and
  the statement is a computer science major is
  called the predicate.
• The statement P(x) x is a computer science
  major, where P denotes the predicate and x is
  not specified is called the propositional
  function P. The value of the propositional
  function at a particular value of x is a
  proposition.
• The statement P(x1,x2, , xn) where (x1,x2,,xn)
  is not specified is also called the propositional
  function P. The value of this propositional
  function at a particular n-tuple (x1,x2,,xn) is
  a proposition. The statement P is also called a
  predicate.
• ?

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Quantifiers

• Universal quantification of P(X) (written ?x
  P(x)) For all x in the Universe of Discourse
  P(x) is true.
• Existential quantification of P(X) (written ?x
  P(x)) There exists an x in the Universe of
  Discourse for which P(x) is true.
• Theorems
• ?(?x P(x)) ? ?x (?P(x) )
• ?(?x P(x)) ? ?x (?P(x) )
• ?

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Other quantifiers

• Uniqueness quantifier (written ?x! P(x)) There
  exists a unique x in the Universe of Discourse
  for which P(x) is true.
• Quantifiers with restricted domains ? xgt0
  P(x)) (for all xgt0 P(x) is true) ? y?0 P(y)
  (there exist a y?0 for which P(y) is true)

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Homework

• Sec 1.3
• pg 46 5, 9, 13, 17, 23, 25, 29, 33, 39, 43

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Sec 1.4

• Nested Quantifiers

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Quantifications of two variables
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Quantifications of two variables and the negation
of the quantifier
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Homework

• Sec 1.4
• pg. 58 3, 5, 8, 10, 14, 19, 25, 33

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Sec 1.5

• Rules of Inference

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Definition

• Argument An argument in propositional logic is
  a sequence of propositions. All but the final
  proposition is called the premises and the final
  proposition is called the conclusion.
• An argument form in propositional logic is a
  sequence of compound proposition involving
  propositional variables. An argument form is
  valid if no matter which particular propositions
  are substituted in its premises, the conclusion
  is true if the premises are all true.
• Note An argument form with premises p1, p2,
  ,pn and conclusion q is valid whenever (p1?p2?
  ?pn ) ? q is a tautology.

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Rules of Inference Simple valid argument forms

• Addition p ? (p ? q)
• Simplification (p ? q) ? p
• Conjunction (p) ? (q) ? (p ? q)
• Modus Ponens p ? (p ? q) ? q
• Modus Tollens ?q ? (p ? q) ? ?p
• HypotheticalSyllogism (p ? q) ? (q ? r) ?
  (p ? r)
• DisjunctiveSyllogism (p ? q) ? ?p ? q
• Resolution (p ? q) ? (?p ? r) ? (q ? r)
• ?

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Fallacies in Proofs

• Affirming the Conclusion (p ? q) ? q ? p
• Faulty conclusions
• Denying the Hypothesis (p ? q) ? ?p ? ?q
  Faulty conclusions
• ?

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Rules of Inference for Quantified Statements
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Homework

• Sec 1.5
• pg 72 3, 5, 7, 9, 13(a,c), 15, 19, 31

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Sec 1.6

• Introduction to Proofs

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Terminology

• Theorem A statement that can be shown to be
  true.
• Axiom (Postulates) Statements that are accepted
  as true without proof.
• Lemma A simple theorem that is used in the
  proof of another theorem.
• Corollary A theorem that can be established
  directly for another theorem.
• Conjecture A statement that is being proposed
  (but not yet proven) to be a true statement.

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Methods of Proving Theorems

• Direct Proofs
• Indirect Proofs
• Vacuous and Trivial Proofs
• Proof by Contradiction
• Proof by Case
• Proofs of Equivalence
• Existence Proofs
• Constructive and Non-constructive
• ?

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Direct Proof

• To Prove that p ? q
• Assume that p is true
• Use rules of inference to show that q is true.
• ?

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Indirect Proof or Proof by Contraposition

• To Prove that p ? q
• Assume that ? q is true
• Use rules of inference to show that ?p is true.
• ?

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Vacuous and Trivial Proofs

• To Prove that p ? q
• Vacuous Proofs Involve false hypothesis
• F ? q is always true, no matter what q is.
• Trivial Proofs involve true conclusions
• P ? T is always true.
• ?

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Proof by Contradiction

• To Prove that proposition p is True
• Show that the assumption that ?p is true leads to
  a contradiction of the form ?p ? (r ? ?r) for
  some proposition r.
• Since the implication ?p ? (r ? ?r) is true, and
  since an implication with a false conclusion can
  only be true if the premise is false, it follows
  that ?p is false and so p must be true.
• ?

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Proof by Contradiction

• To Prove that proposition p ?q is True
• Show that the assumption that ?(p ? r) is true
  leads to a contradiction of the form ?(p ? r)
  ? (r ? ?r) for some proposition r.
• Note here that ?(p ? r) is equivalent to ?(?p ?
  q) which in turn is equivalent to (p ? ?q) .
• So it must be shown that (p ? ?q) ? (r ? ?r)
  for some proposition r.
• ?

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Proofs of Equivalence

• To show p ? q ? r ? y ? z
• Show
• p ? q
• q ? r
• r ?
• y ? z
• z ? p
• ?

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Homework

• Sec 1.6
• pg 86 3, 6, 9, 13, 16, 19, 21, 26, 32

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Sec 1.7

• Proof Methods and Strategies

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Exhaustive Proof or Proof by Case

• (Let the universe of discourse be some finite set
  S.)To prove ?x P(x)
• Prove P(x) is true for each possible value of x
  in S. (only possible when the set of value of x
  is finite)
• ?

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Leveraging Proof by Case

• When attempting to prove a statement by case,
  consider collecting all the cases into disjoint
  classes or set, and then prove that the statement
  is true for all elements in the class. For
  example all integers can be divided into even (a
  mod 2 0) or odd (a mod 2 1) or the can be
  divided into classes based on modular arithmetic
  or based on other integer properties.

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Examples

• Exhaustive
• 1303 is a prime number.
• If x and y are integers than x2 3y2 8 has no
  solutions.
• Leveraging
• If n is a nonzero integer, then n2 gt n.
• If n is an integer than n2 2 is never divisible
  by 4.
• If n is an odd integer than n2 - 1 is always
  divisible by 8.

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Existence Proofs (? x P(x))

• Constructive Existence Proof Show how the value
  x can be constructed with property P(x)
• Existence (non-constructive) Proofs Show that
  there exist an x with the desired properties
  without finding it
• Constructive Existence Example There exist
  perfect numbers. (Numbers equal to the sum of
  their proper divisors.)
• Existence Example There exist irrational
  numbers of the form xy.
• ?

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Uniqueness Proofs

• To Prove There exists an element with a certain
  property
• Proof Requires Two Parts
• Existence Show some such element exists.
• Uniqueness If x and y are two elements with
  this property, show that xy.

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Proof strategies

• Forward Reasoning Start with the hypothesis.
  Use axioms and known theorems construct a
  sequence of steps that leads to the conclusion.
• Backward Reasoning To prove statement q find
  another statement p that can be shown true and
  use it to show that p?q.

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Conjecture

• Fermats Last Theorem (Conjectured some 300 years
  ago) Whenever n gt 2 the equations xn yn zn
  has no integer solutions with xyz ?0. (Solved in
  1990s by Andrew Wilson)
• The 3x1 conjecture Let f(x) x/2 if x is even
  and 3x 1 if x is odd. For ever integer n
  repeated application of f(x) eventually leads to
  1. (Unsolved)

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Homework

• Sec 1.7
• pg 102 3, 7, 10, 14, 22, 27, 28, 37
• End Chapter 1