Mathematics for Computer Science

1
Mathematics for Computer Science

• Lecture 2 Proofs
• Leon van der Torre

2
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
3
Book

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

4
Rehearsal Logic exercise

• In computer chip design, we often use nand
• Using nand and not, find an equivalent expression
  for A and B, A or B, and A implies B

5
A proof?

• Pythagorean proof of a2b2c2.
• The unexpected exam does not exist.
• Every map can be colored with 4 colors so that
  adjacent regions have different colors.
• Every quadratic polynomial over C has two roots
  (by construction).

6
Exercise

• Explain precisely what is wrong with the
  following proof that a cent is equivalent to a
  euro. Which step(s) in the proof is (are) wrong,
  and why?
• 1 cent 0.01 euro (0.1 euro)2
• (10 cent)2 100 cent 1 euro.

7
Methods of ascertaining truth

• Jury Trial Truth is ascertained by twelve people
  selected at random.
• Word of God Truth is ascertained by communication
  with God, perhaps via a third party.
• Word of Boss Truth is ascertained from someone
  with whom it is unwise to disagree.
• Experimental Science The truth is guessed and the
  hypothesis is confirmed or refuted by
  experiments.
• Sampling The truth is obtained by statistical
  analysis of many bits of evidence. For example,
  public opinion is obtained by polling only a
  representative sample.
• Inner Conviction/Mysticism My program is
  perfect. I know this to be true.
• I dont see why not... Claim something is true
  and then shift the burden of proof to anyone who
  disagrees with you.
• Cogito ergo sum Proof by reasoning about
  undefined terms.

8
Euclid, Alexandria, Egypt around 300 BC

• Five assumptions about geometry
• Seemed undeniable based on direct experience.
• E.g., there is a straight line segment between
  every pair of points.
• Propositions that are simply accepted as true are
  called axioms.
• Starting from these axioms, Euclid established
  the truth of many additional propositions by
  providing proofs.
• A proof is a sequence of logical deductions from
  axioms and previously proved statements that
  concludes with the proposition in question.
• Euclids axiom-and-proof approach, now called the
  axiomatic method, is the foundation for
  mathematics.

9
Which axioms?

• Euclidean geometry Given a line l and a point p
  not on l, there is exactly one line through p
  parallel to l.
• Spherical geometry Given a line l and a point p
  not on l, there is no line through p parallel to
  l.
• Hyperbolic geometry Given a line l and a point p
  not on l, there are infinitely many lines through
  p parallel to l.
• Desirable properties consistent and complete.

10
Theorems, lemmas and corollaries

• There are several common terms for a proposition
  that has been proved. The different terms hint at
  the role of the proposition within a larger body
  of work.
• Important propositions are called theorems.
• A lemma is a preliminary proposition useful for
  proving later propositions.
• A corollary is an afterthought, a proposition
  that follows in just a few logical steps from a
  theorem.
• The definitions are not precise. In fact,
  sometimes a good lemma turns out to be far more
  important than the theorem it was originally used
  to prove.

11
Finding proofs

• In principle, a proof can be any sequence of
  logical deductions from axioms and previously
  proved statements that concludes with the
  proposition in question.
• Many proofs follow standard templates.
• Proofs all differ in the details, but these
  templates provide you with an outline to fill in.
  
• Youre free to say things your own way

12
Proving an implication (1)

• In order to prove that P implies Q
• Write, Assume P.
• Show that Q logically follows.
• Example if 0 x 2, then x34x1 gt 0

13
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

14
Good proofs are clear like programs (1)

• State your game plan. E.g. We use case analysis
  or We argue by contradiction.
• Keep a linear flow, a sequential order.
• A proof is an essay, not a calculation. Use
  complete sentences.
• Avoid excessive symbolism. So use words where you
  reasonably can.
• Simplify. Long, complicated proofs take the
  reader more time and effort to understand and can
  more easily conceal errors.

15
Good proofs are clear like programs (2)

• Introduce notation thoughtfully. Sometimes an
  argument can be greatly simplified by introducing
  a variable, devising a special notation, or
  defining a new term (and define the meanings of
  new variables, terms, or notations).
• Structure long proofs. Long programs are usually
  broken into a hierarchy of smaller procedures.
  Long proofs are much the same.
• Dont bully. Words such as clearly and
  obviously serve no logical function.
• Finish. At some point in a proof, youll have
  established all the essential facts you need. Tie
  everything together yourself and explain why the
  original claim follows.

16
Homework

• Read chapter 1 2
• Before next lecture Hand in your answers for
  problem 3 and 4
• (Score will be taken into account for first
  midterm test.)