Proof is the idol before whom the pure mathematician tortures himself.

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• Proof is the idol before whom the pure
  mathematician tortures himself.
• Sir Arthur Eddington
• Programming today is a race between software
  engineers striving to build bigger and better
  idiot-proof programs, and the Universe trying to
  produce bigger and better idiots. So far, the
  Universe is winning.
• Rich Cook

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CSE 502NFundamentals of Computer Science

• Fall 2004
• Lecture 5
• Methods of Proof (Rosen 1.5)
• Proof Strategy (Rosen 3.1)

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Rules of Inference

• Rules that provide justification of the steps
  used to show that a conclusion follows logically
  from a set of hypotheses
• Addition p is true therefore p Ú q is true
• Simplification p Ù q is true therefore p is
  true
• Conjunction p is true, q is true therefore p Ù
  q is true
• Modus ponens p is true, p q is true therefore
  q is true
• Modus tollens Øq is true, p q is true
  therefore Øp is true
• Hypothetical syllogism p q is true, q r is
  true therefore p r is true
• Disjunctive syllogism p Ú q is true, Øp is true
  therefore q is true
• Resolution p Ú q is true, Øp Ú r is true
  therefore q Ú r is true

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Valid Arguments

• An argument is valid if whenever all the
  hypotheses are true, then the conclusion is true
• (p1 Ù p2 Ù Ù pn) q
• A valid argument can lead to an incorrect
  conclusion if one or more false propositions are
  used
• Show that the following hypotheses lead to the
  conclusion It rained.
• If it does not rain and if it is not foggy, then
  the sailing race will be held and the lifesaving
  demonstration will go on.
• If the sailing race is held, then the trophy
  will be awarded.
• The trophy was not awarded.

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Common Fallacies

• Fallacy of affirming the conclusion
• (p q) Ù q p
• If you practice ten hours per day, then you will
  learn to play the flute. You learned to play the
  flute.
• Therefore, you practiced ten hours per day.
• Fallacy of denying the hypothesis
• (p q) Ù Øp Øq
• If you practice ten hours per day, then you will
  learn to play the flute. You did not practice
  ten hours per day.
• Therefore, you did not learn how to play the
  flute.

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Predicates

• x 3 is not a proposition
• x is a variable, is greater than 3 is a
  predicate
• P(x) x 3 is a propositional function of x
• P(4) is true
• P(3) is false
• P(x,y) x y 3 is a propositional function of
  x and y
• P(1,4) is false
• P(2,1) is true

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Quantifiers

• Universal quantification "xP(x)
• P(x) is true for all values of x in the universe
  of discourse
• Example let P(x) be x 1 x and let the
  universe of discourse be all real numbers, then
  "xP(x) is true
• When all elements of the universe of discourse
  can be counted, x1 xn, "xP(x) is the same as
  P(x1) Ù Ù P(xn)
• To show that "xP(x) is false, we need to find
  only one value of x in the universe of discourse
  for which P(x) is false (counterexample)
• Existential quantification xP(x)
• There exists a value of x in the universe of
  discourse such that P(x) is true
• Example let P(x) be x 3 and let the universe
  of discourse be all real numbers, then xP(x) is
  true
• When all elements of the universe of discourse
  can be counted, x1 xn, xP(x) is the same as
  P(x1) Ú Ú P(xn)
• To show that xP(x) is true, we need to find
  only one value of x in the universe of discourse
  for which P(x) is true

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Rules of Inference for Quantifiers

• Universal instantiation "xP(x) is true,
  therefore P(c) is true where c is a particular
  member of the universe of discourse
• Universal generalization P(c) is true where c is
  an arbitrary member of the universe of discourse,
  therefore "xP(x) is true
• Existential instantiation xP(x) is true,
  therefore P(c) is true for some element c
• Existential generalization P(c) is true for some
  element c, therefore xP(x) is true

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Proving Theorems

• Many theorems are implications, thus implication
  proof techniques are important
• p q is true unless p is true and q is false
• Direct proofs prove p q is true by showing
  that if p is true, then q must also be true
• Indirect proofs prove p q is true by showing
  that contrapositive, Øq Øp, is true
• Vacuous proof prove p q is true by showing
  that the hypothesis p is false
• Trivial proof prove p q is true by showing
  that the conclusion q is true

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Other proof techniques

• Proof by contradiction
• Find a contradiction q such that Øp q is true
  (Øp F is true)
• Thus, the proposition Øp must be false and p must
  be true
• Example Show that at least four of any 22 days
  must fall on the same day of the week
• Proof by cases
• Prove an implication of the form (p1 Ú p2 Ú Ú
  pn) q by proving each of the implications pi
  q for i 1n
• (p1 Ú Ú pn) q (p1 q) Ù Ù (pn q)
• Often convenient to prove p q by using a
  disjuction p1 Ú p2 Ú Ú pn º p
• Proofs of equivalence
• Prove a biconditional p q using the tautology
  (p q) (p q) Ù (q p)

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Proofs with Quantifiers

• Existence proofs prove theorems that are
  assertions that objects of a particular type
  exist
• Theorem is a proposition of the form xP(x)
• Constructive existence proof prove xP(x) is
  true by finding an element c such that P(c) is
  true
• Nonconstructive existence proof prove xP(x) is
  true by showing that the negation of the
  existential quantification implies a
  contradiction
• Uniqueness proofs prove theorems that assert the
  existences of a unique element with a particular
  property
• Existence how that an element with this property
  exists
• Uniqueness how that no other element has this
  property
• x(P(x) Ù "y(y¹x ØP(y)))
• Counterexample proofs prove theorems of the form
  "xP(x) are false by finding a counterexample