Mathematics

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Mathematics

• numbers variables functions logic proof

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Reflection on irrationalityand the need for proof
No matter how fine the sub-divisions on a ruler
that precisely measures the length of the side of
a square, the diagonal will not be measured in a
whole number of units by that ruler. And any
ruler that measures the diagonal will not measure
the sides. The proof of this destroyed the
Pythagorian Brotherhood and separated geometry
and algebra for a thousand years.
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A proof of Pythagorass Theorem
Euclid tried to write down a complete list of
axioms that were assumed to be true for geometric
proofs. (But flat geometry is not the only
geometry.)
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The angles of a triangle sum to two right angles
The hypothesis that parallel lines are cut by a
transversal at the same angle is called the
parallel postulate.

• Draw lines perpendicular to BC at A, B and C.
• From the parallel postulate it is clear that the
  sum of the angles is two right angles.

So, if the parallel postulate is accepted then
the sum is 180
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Direct proof

• A direct proof begins by assuming that some
  particular hypotheses are true and producing a
  sequence of propositions equivalent to all or
  some of these axioms.
• If we accept the hypotheses as true, then the
  proposition we finish with, the conclusion, must
  also be accepted as true it is proved.
• A direct proof is a valid logical argument
  when the premises are true, the conclusion is
  true.

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The square of any even number is even

• Proof
• If a integer is even, it means that it is
  divisible by 2, which means that, if the number
  is n, then ? an integer k such that n 2 k.
  Squaring both sides of this equality gives
  another equality n2 2 k2 .
• This shows that the number n2 is divisible by
  2 because the result of the division is an
  integer k2 . Consequently n2 is an even
  number.Since n is any integer, we can say that
  the square of any integer is even.
• (Italian mathematician Giuseppe Peano tried to
  write down all the axioms that were basic for
  doing arithmetic.)

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If n2 is even is n necessarily even?

• This is the converse proposition.
• The contrapositive of m2 even ? m even is
  m not even ? m2 not even or m odd ?
  m2 odd
• and this is easy to show directly since m
  2k 1 ? m2 4k2 4k 1
• ? m2 2 (2k2 2k) 1
• ? m2 2 K 1 where K is an integer.

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(Indirect) Proof by Contradictionof the
irrationality of ? 2

• We will show that the proposition p ? ?2 is
  irrational ( that is, ?2 not a fraction)
• is true by showing that p must be false.
• If p were true, that is ?2 is a rational
  number, then there would have to be two whole
  numbers m and n such that ?2 m / n ?
  m2 2 n2 (squaring both numbers and
  multiplying both by n2)? m2 is even
• ? m is even (a fact we just proved)

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Continuing proof of irrationality of ? 2

• ?2 m /n ? m2 2 n2 ? m is even.
• If m is even then m 2 k and 4 k2 2 n2 ? n2
  2 k2 ? n is even.
• Dividing both m and n by 2, we start again with
  ?2 m1 / n1 and again prove that m1 and n1
  are both even.
• So, after r repetitions m and n are
  divisible by 2r for any whole number r and
  hence larger than any number we can think of.
  No finite whole number can have this
  property.Thus the statement that ?2 is rational
  is logically equivalent to a contradiction.
  Therefore the negation is true ?2 is
  irrational.

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Methods of proof

• In the preceding examples we saw three methods of
  proof
• Direct proof
• (Direct) proof through the contrapositive
• (Indirect) proof by contradiction (proving the
  negation is false)

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Example (Proof by Contradiction)

• The number of prime numbers is infinite. (p)
  If the number of primes is not infinite (?p),
  then there are a finite number of primes they
  can be listed and so q all primes are in
  the set S r1, r2, r3, r4, , rn The
  number r r1? r2 ? r3 ? r4 ? ? rn 1 is
  prime since it's remainder when divided by any
  one of the primes listed is 1 r is not in the
  set S since its bigger than any of one them.
  So, S is not the set of all primes (?q ). This
  is a contradiction
• q ? ?q

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The universal membership function, ?

• In symbolic algebra, symbols, such as S and x,
  are used to represent sets and their elements.
• The symbol,? ?, is used to represent the phrase
  (is) an element of so that
  x ? S is short for x is an element of S.
  
• From a computing perspective x ? S would be
  the syntax for an in-fix function, ? returning 0
  or 1. In J it is e. and checks whether the
  left argument is one of the elements of the right
  argument.
• We will use it as a generic propositional
  function for membership even when we dont have
  an algorithm for it.

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The universal quantifier ? for all for
any

• The statement all rats are gray applies to the
  set, R, of rats and it is a proposition. If x
  is a rat, i.e. x ? R is true, then the
  proposition is that the statement is true
  whatever x is chosen. We write ? x ( x ? R
  ) ? ( G(x) 1 ) or, for short, simply
  ? x G(x)where G is the propositional
  function for the predicate, is gray, (ie. G is
  a method for determining whether any particular x
  is gray or not.)
• It only take one rat to fail this test for the
  proposition to be false.

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The existential quantifier ? for some there
is there exists

• The statement, some rats are gray, implies that
  in amongst the rats there is at least one that is
  gray (or an element of the set of gray things.
  We could write ? x (x ? R ) ? (x ? G)
  is true where G is the set of gray things, or,
  for short ? x R(x) ? G(x) where
  R(x) determines x ? R and G(x) determines if x
  ? G. Assuming that the universe of discourse
  is the set R we could write ? x G(x)
  (read as such that)
• It only take one rat to pass this test for the
  proposition to be true.

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Prove that for some by example

• A statement of the form
• ? n P(n)
• can be proved true by finding just one case for
  which P(n) is true.
• Pr some mammals live in the sea.
• T since Whales are mammals and live in the sea.
• Pr 19x4 -26x3 7 lt 0 (for some x)
• T since 19(1.024)-26(1.023)7 -0.0252 (in
  fact for any x in 1 lt x lt 1.051)

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Proofs for all cannot rest on examples

• Proving that a predicate is true for all ?n
  P(n)
• or that a predicate is true for none, ie. not
  true for any ?
  (?n P(n))
• cannot be be done by an example.
• Examples can only show that Pr is true, or not
  true, for some.
• To prove that a predicate is true for all
  subjects requires a demonstration for any subject
  a demonstration that is independent of the
  subject.

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Algebraic variables

• Direct proofs for any and every element of a set
  rest on working algebraically with an arbitrary
  element of the set represented by an algebraic
  variable i.e. a variable whose value is not
  assigned. The argument should remain valid no
  matter what value might be assigned to the
  variable.
• Assigned variables in a computer represent
  either the complete set of values (for a finite
  set) or a sample of values from an (infinite) set
  that is adequate for a faithful representation of
  behaviour.
• Notice that algorithms are always written in
  terms of algebraic variables whose values are
  assigned at a function call.

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Lists of sample values from a set

• When we deal with predicates on a computer, we
  can take all or a sample of values from the
  universal set and test the proposition on the
  sample to get some idea of the subset for which
  the proposition is true.
• x ? -1, 0, 1, 2, 3, 4, 5, 6, 7
• x gt 2 0 0 0 0 1 1 1 1 1
• x lt 6 1 1 1 1 1 1 1 0 0
• (xgt2)?(xlt6) 0 0 0 0 1 1 1 0 0
• Predicates are just relationship functions with
  the input left unassigned.

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Checking propositions
J code
x . _4 i. 9 NB. _4 to 4y . . i. 17
NB. 16 to 0test 3 '( y) gt ( .
y )2 test each x , each / y
y (? each) x2 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1
1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1
1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0
1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0
0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1
0 0 0 0 0 0 0 1 0 0 0 0
y ? x2
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Direct arguments obtainable through the converse

• Direct arguments rely on accepted hypotheses, h,
  which we regard as self-evident truths and a
  chain
• h ? p p ? . . . ? q q ? r
• Thus if h is true then r is true. When we don't
  know which h to start with we try
• r ? s s ? . . . ? t t ? hwhere the
  implications, we hope, might be reversible. But
  we have to test them.

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Algebraic problems are often solved by converse
arguments

• For example, what size should a square tin box be
  if it's sides must be 4cm high and the amount of
  tin used should be 57 sq. cm.
• Here we require the conclusion that whatever the
  side of the box (say x) x216x 57.
• We begin with the conclusion. If x216x 57
  then (x8)2121 and so x -19 or 3. We must now
  show that if x 3 then x216x 57.
• It does not follow that there is a box of side
  -19

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Answers from converse reasoning must be tested

• 4 ?(x 3) x 1
• ?(x 3) x 3
• x 3 x2 6x 9 (1)
• x2 7x 6 0
• x 1 or 6
• x 6 works ! but x 1 does not
• Equation (1) could also have come from
• ?(x 3) 3 x or 4 ?(x 3) 7
  x

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Comparing geometric and arithmetic means

• ?(ab) 1 2 3 4 5
  6 7 8 9 10
• 1 1.00 1.41 1.73 2.00 2.24 2.45
  2.65 2.83 3.00 3.16
• 2 1.41 2.00 2.45 2.83 3.16 3.46
  3.74 4.00 4.24 4.47
• 3 1.73 2.45 3.00 3.46 3.87 4.24
  4.58 4.90 5.20 5.48
• 4 2.00 2.83 3.46 4.00 4.47 4.90
  5.29 5.66 6.00 6.32
• 5 2.24 3.16 3.87 4.47 5.00 5.48
  5.92 6.32 6.71 7.07
• 6 2.45 3.46 4.24 4.90 5.48 6.00
  6.48 6.93 7.35 7.75
• (a b)/2
• 1 1.00 1.50 2.00 2.50 3.00 3.50
  4.00 4.50 5.00 5.50
• 2 1.50 2.00 2.50 3.00 3.50 4.00
  4.50 5.00 5.50 6.00
• 3 2.00 2.50 3.00 3.50 4.00 4.50
  5.00 5.50 6.00 6.50
• 4 2.50 3.00 3.50 4.00 4.50 5.00
  5.50 6.00 6.50 7.00
• 5 3.00 3.50 4.00 4.50 5.00 5.50
  6.00 6.50 7.00 7.50
• 6 3.50 4.00 4.50 5.00 5.50 6.00
  6.50 7.00 7.50 8.00

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Typical graph of points where geometric and
arithmetic means are equal
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Proof that geometric mean is less than the
arithmetic mean

• Given two quantities a and b, is the geometric
  mean ?(ab) larger or smaller than the
  arithmetic mean (a b)/2.
• Supposing we assume that ?(ab) gt (a b)/2,it
  would follow that 4a b gt (a b)2 or that4a b gt
  a 2 2ab b2 ie. 0 gt a 2 - 2ab b2 (a - b)2
  lt 0 which is false !! Therefore its negation
  is true.
• If we can reverse the argument, we also have a
  proof.

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?(ab) lt (a b)/2

• It is always true that (a - b)2 gt 0 , that is
  thata 2 - 2ab b2 gt 0 from which it follows
  that a 2 2ab b2 gt 4ab (adding 4ab to
  both sides).
• Thus we have (a b)2 gt 4ab
• and, taking square roots, that a b gt 2
  ?(ab)
• and, if a and b are both positive, then(a b)
  /2 gt ?(ab)
• If a and b are both negative then (a b) /2 lt
  -?(ab).

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Conclusions a fortiori

• A deduction from a stronger result is said to be
  a deduction from the stronger or a deduction a
  fortiori.
• It is true (can you prove it?) that 12 22
  32 42 52 n2 n(n1)(2n1)/6

It follows that at least one of n, n1 and 2n1
is divisible by 3 and of course either n or n1
is divisible by 2.
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Whats the big idea?

• There is need for proof in mathematics. It is
  based on axioms and logical deduction.
• Direct and indirect proofs and using the
  contrapositive. Proofs by example only work for
  quantified predicates ? x P(x) can be proved
  by providing just one x?x P(x) can be
  disproved by providing just one x.
• Using reversible arguments (equivalent
  propositions) to produce a valid deductive
  sequence.