Mathematics for Computer Science

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Mathematics for Computer Science

• Lecture 3 More proofs
• Leon van der Torre

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Slides

• Slides can be found on the internet
• These slides are based on MIT open courseware,
  by S. Devadas and E. Lehmann

BECS - Bachelor of Engineering in Computer
Science
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Book

• Rod Haggarty,
• Discrete Mathematics for Computing
• Diskrete Mathematik für Informatiker

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Additional reading

• Rosen, Kenneth H. Discrete Mathematics and its
  Applications, Fourth Edition, McGraw-Hill, Inc.,
  New York, 1999.
• Velleman, Daniel J. How to prove it A Structured
  Approach, Cambridge University Press, Cambridge,
  England, 1994.

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Tautologies

• P P is a tautology P is always true
• (P ? Q) ? (Q ? P)
• P ? (Q ? R) ? (P ? Q) ? R
• P ? (Q ? R) ? (P ? Q) ? (P ? R)
• (P ? P) ? P
• ??P ? P
• ? (P ? Q) ? ?P ? ?Q

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Rehearsal Are these saying the same?

• If I am hungry (P) , then I am grumpy (Q).
• If I am not grumpy, then I am not hungry.
• (P ? Q) ? (?Q ? ?P)

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Rehearsal are these saying the same?

• If I am hungry (P) , then I am grumpy (Q).
• If I am grumpy, then I am hungry.
• not (P ? Q) ? (Q ? P)

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Rehearsal proving an implication (2)

• Write, We prove the contrapositive and then
  state the contrapositive.
• Proceed as in Method 1.
• Example If r is irrational, then vr is also
  irrational.
• Exercise If m2 is even, then m is even
• (P ? Q) ? (?Q ? ?P) but not (P ? Q) ? (Q ? P)

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Tautologies represent proof methods

• (P ? Q) ? (?P ? Q)
• (P ? Q) ? (?Q ? ?P)
• ? (P ? Q) ? (P ? ?Q)
• (P ? Q ? R) ? (P ? ?Q ? R)
• (P ? Q ? R) ? (P ? R) ? (Q ? R)
• (P ? Q ? R) ? (P ? Q) ? (P ? R)

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Reasoning backward?

• (ab)/2 vab
• ab 2vab
• (ab)2 4ab
• a22abb2 4ab
• a2-2abb2 0
• (a-b)2 0
• Whats wrong with this proof?

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Rehearsal If and only if

• P if and only if Q P and Q logically
  equivalent
• That is, either both are true or both are false.
• x2 - 4  0 if and only if x 2
• Doesnt arise in ordinary speech, abbreviated
  iff.

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Rehearsal Are these saying the same?

• These two statements together
• If I am grumpy, then I am hungry.
• If I am hungry, then I am grumpy.
• I am grumpy if and only if I am hungry.
• (P ? Q) ? (P ? Q) ? (Q ? P)

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Proving an If and Only If (1)

• Prove each statement implies the other
• Write, We prove P implies Q and vice-versa.
• Write, First, we show P implies Q. Do this by
  one of the methods discussed last week.
• Write, Now, we show Q implies P. Again, do this
  by one of the methods.
• Example Let A, B and C be sets, then
• A ? (B ? C) (A ? B) ? (A ? C)

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Proving an If and Only If (2)

• Construct a Chain of Iffs
• Write, We construct a chain of if-and-only-if
  implications.
• Prove P is equivalent to a second statement which
  is equivalent to a third statement and so forth
  until you reach Q.
• Example The standard deviation of a sequence of
  values is 0 iff all values are equal to the mean

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More tautologies ( proof methods)

• P P is a tautology P is always true
• ?P ? (P ? Q)
• P ? (P ? Q) ? Q
• (P ? Q) ? ?Q ? ?P
• (P ? Q) ? (Q ? R) ? (P ? R)
• (P ? Q ) ? (?P ? R) ? Q ? R

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Proof by contradiction

• (?P ? Q) ? ?Q ? P when Q is contradiction
• To prove a proposition P by contradiction
• Write, We use proof by contradiction.
• Write, Suppose P is false.
• Deduce a logical contradiction.
• Write, This is a contradiction. Therefore,
  P must be true.
• Example. v2 is irrational.
• Example. primes is infinite

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Proof by case analysis

• (P ? R) ? (Q ? R) ? (P ? Q ? R)
• Write, We use case analysis
• Identify a sequence of conditions, at least one
  of which must hold.
• For each condition
• State the condition
• Prove P assuming that the condition holds
• Example there exist irrational numbers p and q
  such that p to power q is rational.

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Proofs in programming language Prolog

• Represent formulas as sets of clauses
• Remove implications
• Move ? directly in front of variables
• Use distribution of ? over ?
• Use resolution to derive contradiction
• (P ? Q ) ? (?P ? R) ? Q ? R

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Conversion to CNF

• B1,1 ? (P1,2 ? P2,1)
• Eliminate ?, replacing a ? ß with (a ? ß)?(ß ?
  a).
  
• (B1,1 ? (P1,2 ? P2,1)) ? ((P1,2 ? P2,1) ? B1,1)
• 2. Eliminate ?, replacing a ? ß with ?a? ß.
• (?B1,1 ? P1,2 ? P2,1) ? (?(P1,2 ? P2,1) ? B1,1)
• 3. Move ? inwards using de Morgan's rules and
  double-negation
  
• (?B1,1 ? P1,2 ? P2,1) ? ((?P1,2 ? ?P2,1) ? B1,1)
• 4. Apply distributivity law (? over ?) and
  flatten
  
• (?B1,1 ? P1,2 ? P2,1) ? (?P1,2 ? B1,1) ? (?P2,1 ?
  B1,1)
  

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Resolution example

• KB (B1,1 ? (P1,2? P2,1)) ?? B1,1
• a ?P1,2

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Homework

• Read chapter 1 2
• Before next Monday Hand in your answers for
  problem 5 and 6
• (Score will be taken into account for first
  midterm test.)