Proofs, Wonderful Proofs by D'N' SeppalaHoltzman St' Josephs College Brooklyn, NY

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Proofs, Wonderful ProofsbyD.N.
Seppala-HoltzmanSt. Josephs CollegeBrooklyn,
NY
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Genuine Proofs Only in Mathematics

• No other branch of academic inquiry has the
  notion of genuine proof.
• Many others use the word proof but what they
  mean by it is overwhelming evidence.
• Science, law and most other realms of human
  inquiry use inductive reasoning to establish what
  they call proof.
• Mathematics uses deductive reasoning.

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Inductive Reasoning

• In inductive reasoning, one observes the real
  world and imagines rules that govern the
  observed reality.
• These rules are then tested by experiments that
  either confirm the imagined rules or refute them.
• If the system is refuted by contradictory
  evidence, it is either abandoned or refined.

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Inductive Reasoning

• If empirical evidence refutes an entire theory,
  it is abandoned.
• If a refinement is required, the smallest
  possible changes are made in the rules that make
  the system compatible with the newly observed
  evidence.
• The refined system is now repeatedly tested for
  possible corroborations or refutations.

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Inductive Reasoning

• If corroborating evidence becomes overwhelming,
  the theory is relabeled a truth and is accepted
  as such.
• The evidence that led up to this point is
  considered proof of the accepted truth.
• Whatever the likelihood, new evidence is always
  possible that may contradict parts or even all of
  an accepted truth.

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Deductive Reasoning

• Mathematics uses deductive reasoning.
• In deductive reasoning, one uses an axiomatic
  system.
• Some number of axioms (or assumed, unproven but
  accepted truths) is given.
• Some set of rules is presented that make precise
  what it means for one statement to follow
  logically from another.

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Deductive Reasoning

• If one were to accept the axioms and rules of
  inference of a given mathematical system, any
  logically valid proof within that system would
  yield an irrefutable, undeniable, everlasting
  truth in that system.

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Proof

• Given some axiomatic system, a proof that some
  conclusion follows from a set of hypotheses is a
  sequence of statements that make the conclusion
  an undeniable, logical consequence of the
  hypotheses.
• Typically, one begins with the given hypotheses
  and makes a sequence of statements, each
  following from the previous ones, until the
  conclusion is reached and is irrefutable.

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Proof

• Proofs need not start with the hypotheses but
  they must make use of them somewhere.
• In any event, one starts with a universally
  agreed upon truth and then proceeds to show that
  some other statement follows logically from that
  statement.
• A sequence of steps of this sort is made until a
  an undeniable conclusion is reached.

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Indirect Proof

• One particularly powerful method of proof is
  called indirect proof.
• Here one asks what would go wrong if the
  conclusion were not to follow from the
  hypotheses.
• The method begins by assuming the negation of
  what one wanted to conclude and proceeds to
  deduce some logical absurdity as a consequence of
  this assumption.
• If a contradiction results from our assumption,
  it must have been a false assumption. Thus, the
  desired conclusion must have been true, after
  all.

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Proof vs. Truth

• There are pros and cons to both deductive and
  inductive reasoning.
• In the scientific method, inductive reasoning
  converges on the truth of some real, physical
  entity.
• It never actually achieves absolute certainty
  (nothing can in the real world) but it can come
  very close, indeed.

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Proof vs. Truth

• Mathematical proof, on the other hand, does
  achieve absolute certainty.
• Alas, the truth that one has established is not a
  truth in the real world.
• Mathematical models of the real world are always
  imprecise.
• Thus, neither science nor mathematics can provide
  us with absolute truth about the real world.

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Proofs are Wonderful!

• Providing truth about the real world is not the
  only measurement of value.
• It is often the case that one seeks to comprehend
  something that appears, initially, to be
  impenetrable, opaque, confounding.
• The shear joy of the Aha! experience, when
  proof illuminates the darkness, is nothing short
  of wonderful!

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Wonderful Proofs

• To illustrate this, a small number of wonderful
  proofs have been selected.
• The following selection criteria were used
• The initial response should be, How could
  anybody possibly prove this?!
• The proof should be short, simple and beautiful.
• If possible, the result should also be a
  significant result in mathematics.

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1) The Number of Primes is Infinite

• Recall that a prime number is a positive integer
  that is divisible only by itself and 1.
• Euclid proved that there are infinitely many by
  indirect proof. That is, he assumed that there
  were only finitely many P1 P2 P3 Pk

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Infinite Number of Primes

• Consider N (P1 x P2 x x Pk) 1
• Thus, N is one more than the product of ALL of
  the primes.
• N, itself, must be either a prime or a product of
  primes.
• If N is a prime, it clearly is not one on our
  list.
• If it is a composite, it is not divisible by any
  prime on our list as it leaves a remainder of 1
  when divided by any of them.

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Infinite Number of Primes

• In either event, we have determined that there
  must be some additional prime number that was not
  on our list.
• But this list was assumed to be of ALL the
  primes.
• This contradiction proves the result.

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2) The Real Numbers are Uncountable

• An infinite set is called countable when its
  members can be placed in one-to-one
  correspondence with the natural numbers
  1,2,3,
• The integers and the rational numbers are
  countable sets.
• Cantor proved that the real numbers in the open
  interval from 0 to 1 is uncountable, i.e. a
  bigger infinity.

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Uncountable Reals

• Once again, we proceed by indirect proof and
  suppose that the reals in the unit interval are
  countable.
• If this set is countable, then there exists some
  one-to-one correspondence between ALL of its
  members and the set of natural numbers.

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Uncountable Reals

• Suppose that our correspondence looked like this
  
• 1 lt-gt 0.3487309..
• 2 lt-gt 0.4283045..
• 3 lt-gt 0.9342650..
• 4 lt-gt 0.5000000..
• etc.

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Uncountable Reals

• Let us now create a real number in the unit
  interval B 0.b1b2b3
• b1 is taken to be any integer other than 0, 9 or
  the first digit of the real number that
  corresponded to 1 on our list.
• b2 is taken to be any integer other than 0, 9 or
  the second digit of the real number that
  corresponded to 2 on our list.
• Etc.

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Uncountable Reals

• The digits 0 and 9 were avoided to prevent
  confusion resulting from dual representations
  like 0.50000.. 0.49999..
• Now B is a real number in the unit interval but
  it is not anywhere on our list (which was
  supposed to be complete) since it differs from
  the jth entry in the jth place.
• This contradiction proves the result.

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3) The Rationals are Countable

• 0
• 1/1 -1/1 2/1 -2/1 3/1
• ½ -1/2 2/2 -2/2 3/2
• 1/3 -1/3 2/3 -2/3 3/3
• ..

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Countable Rationals

• Here we have listed all of the rationals (with
  repetitions) and have provided a way to count
  them by defining a 1st, 2nd, 3rd, and so on.
• When we reach a number already counted, we simply
  skip over it.

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Corollary

• There are many more irrational numbers than
  rational ones, i.e. a larger infinity.
• This follows from the fact that the real numbers
  is the union of the sets of irrational numbers
  and rational numbers.
• Since the union of two countable sets is
  countable, the set of irrationals must be
  uncountable since the rationals were countable.

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A Diagram Worth 1,000 Words

• Sometimes a diagram is all that is required for a
  proof to become evident.
• Consider the following proof of the Pythagorean
  Theorem.
• Here, we take a square (ab) on a side, and
  divide it up into 4 a-b-c right triangles and a
  central c x c square.

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4) Pythagorean Theorem

• a2 b2 c2

a
b
b
a
c
c
c
a
c
b
a
b
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Pythagorean Theorem

• (a b)2 a2 b2 2ab area of square
• 4 x ½ (a x b) 2ab area of 4 triangles
• c2 area of central square
• Thus c2 2ab a2 b2 2ab area of big
  square.
• Subtracting 2ab from both sides gives the desired
  result.

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Wow!

• These are but a very few illustrations of the
  beauty and power of mathematical proof.
• In each case, we began with a seemingly
  impossible task and asked How on Earth can one
  prove this?!
• Moments later we had an incontrovertible argument
  before us that demonstrated a universal, timeless
  truth.