Matrices Proof Strategy Sequences and Summations

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MatricesProof StrategySequences and Summations
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Learning Objectives

• Matrix definitionMatrix arithmeticMatrix
  properties.
• Proof strategy.
• Sequences and summations.

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Matrix Definition

• A matrix is a rectangular array of numbers.
• A matrix with m rows and n columns is called an m
  x n matrix.
• Square matrix has the same number of rows and
  columns.
• Two matrices are equal if they have the same
  number of rows, the same number of columns, and
  the same elements.

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Matrix Definition

• A matrix is also an application from N x N to a
  set E, generally R.
• (i,j) ? aij
• ith row is (ai1, ai2 , , ain )
• jth column is (a1j, a2j , , anj )

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Matrix Definition

• procedure matrixEquality(A, B matrices)
  equal true for i1 to m for
  j1 to n if not aij bij
  then equal false
  equal is true if A and B are equal, false
  otherwise

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Matrix Arithmetic

• Addition let Aaij and B bij two matrices m
  x n, then the matrix AB aij bij and is also
  m x n.

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Matrix Arithmetic

• Matrix addition algorithmprocedure
  matrixAddition(A, B matrices) for i1 to m
  for j1 to n cij
  aij bij Ccij is the sum of A and B
  

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Matrix Arithmetic

• Multiplication let Aaij an m x k matrix, and
  B bij a k x n matrix. Then the matrix AB
  cij , product of A by B, is m x n, and is
  defined as

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Matrix Arithmetic

• Matrix product algorithmprocedure
  matrixProduct(A, B matrices) for i1 to m
  for j1 to n begin
  cij 0 for
  q 1 to k cij cij
  aiqbqj end Ccij is
  the sum of A and B

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Matrix Properties

• The product of two matrices is not commutative.
• First of all, the two matrices need to be square.
• Even in this case, in general AB ? BA.
• The product of matrices is associative(AB)C
  A(BC)
• The product of matrices is like composition of
  functions. It represents the composition of
  functions between vector spaces.

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Matrix Properties

• The identity matrix is defined as a square matrix
  with all zeros, except on the diagonal, which
  contains only 1.
• In ?ij were ?ij 1 if i j,
  and 0 otherwise.

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Matrix Properties

• In is the neutral element (zero) of matrix
  multiplication for square matrices of order n.
• InAAIn A
• Power AAA An _________
  n times

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Matrix Properties

• The transpose of a matrix is obtained by
  exchanging the lines and columns.

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Matrix Properties

• A symmetric matrix is a matrix that is equal to
  its transpose. It is a square matrix.

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Matrix Properties

• Matrices are used in computer science to
  represent documents (information retrieval).
• A collection of n texts and m different words can
  be represented by an m x n matrix, where each
  document is a line and each word is a column. If
  a document contains a word, then the
  corresponding matrix element is set to 1,
  otherwise it is set to 0.
• The result is a zero-one matrix that is sparse
  mostly containing zeros. Specific methods in data
  structures are used to represent sparse matrices.

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Proof Strategy

• There are many proof strategies
• Forward chaining (from hypotheses to conclusion)
• Backward chaining (from conclusion to hypotheses)
• Proof by cases (odd numbers, even numbers)
• Adapting existing proofs
• Conjecture and proof
• Counterexamples
• The halting problem

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The proof, using proof by cases...

• Given ngt0, prove there is a prime pgtn.
• Consider x n!1. Since xgt1, we know (x is
  prime)?(x is composite).
• Case 1 x is prime. Obviously xgtn, so let px
  and were done.
• Case 2 x has a prime factor p. But if p?n, then
  p mod x 1. So pgtn, and were done.

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The Halting Problem (Turing36)

• The halting problem was the first mathematical
  function proven to have no algorithm that
  computes it!
• We say, it is uncomputable.
• The desired function is Halts(P,I) the truth
  value of this statement
• Program P, given input I, eventually
  terminates.
• Theorem Halts is uncomputable!
• I.e., There does not exist any algorithm A that
  computes Halts correctly for all possible
  inputs.
• Its proof is thus a non-existence proof.
• Corollary General impossibility of predictive
  analysis of arbitrary computer programs.

Alan Turing1912-1954
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The Proof

• Given any arbitrary program H(P,I),
• Consider algorithm Breaker, defined asprocedure
  Breaker(P a program) halts H(P,P) if halts
  then while T begin end
• Note that Breaker(Breaker) halts iff
  H(Breaker,Breaker) F.
• So H does not compute the function Halts!

Breaker makes a liar out of H, by doing the
opposite of whatever H predicts.
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Goldbachs Conjecture

• Some very simple statements of number theory
  havent been proved or disproved!
• E.g. Goldbachs conjecture Every integer n2 is
  exactly the average of some two primes.
• ?n2 ? primes p,q n(pq)/2.
• There are true statements of number theory (or
  any sufficiently powerful system) that can never
  be proved (or disproved) (Gödel).

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Sequences and Summations

• Definition Two sets A and B have the same
  cardinality if there is a bijection f A? B.
• The cardinality of A will be denoted by A .
• Two finite sets have the same cardinality if they
  have the same number of elements.
• N and N x N have the same cardinality. N and N x
  N x N also have the same cardinality.
• Definition An infinite set A is countable if A
  N

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Sequences and Summations

• Definition A sequence is a function from a
  subset of the natural numbers (usually of the
  form 0, 1, 2, . . . to a set S.
• Note the sets 0, 1, 2, 3, . . . , k and 1, 2,
  3, 4, . . . , k are called initial segments of
  N.
• Notation if f is a function from 0, 1, 2, . .
  . to S we usually denote f(i) by ai and we write
• where k is the upper limit (usually ?).

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Sequences and Summations

• Examples Using zero-origin indexing, if f(i)
  1/(i 1). then the sequence f 1, 1/2,1/3,1/4,
  . . . a0, a1, a2, a3, Using one-origin
  indexing the sequence f becomes 1/2, 1/3, . . .
  a1, a2, a3, . . .

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Sequences and Summations
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Sequences and Summations

• Product notation
• Definition A geometric progression is a sequence
  of the form a, ar, ar 2 , ar 3 , ar 4 , . . .
• Your book has a proof that

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Sequences and Summations

• Proof by induction reasoning
• property is true for n0
• Hypothesis true for all k lt n
• Let us take n1.
• You should also be able to determine the sum
• if the index starts at k vs. 0
• if the index ends at something other than n
  (e.g., n-1, n1, etc.).

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Sequences and Summations

• CardinalityDefinition The cardinality of a set
  A is equal to the cardinality of a set B, denoted
  A B , if there exists a bijection from A
  to B.
• Definition If a set has the same cardinality as
  a subset of the natural numbers N, then the set
  is called countable.If A N, the set A is
  countably infinite.
• The (transfinite) cardinal number of the set N is
  aleph null.
• If a set is not countable we say it is
  uncountable.

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Sequences and Summations

• ExamplesThe following sets are uncountable (we
  show later)
• The real numbers in 0, 1
• P(N), the power set of N
• Note With infinite sets proper subsets can have
  the same cardinality. This cannot happen with
  finite sets.
• Countability carries with it the implication that
  there is a listing of the elements of the set.

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Sequences and Summations

• Definition A ? B if there is an
  injection from A to B.
• ExampleTheorem If A is a subset of B then A
  ? B .Proof the function f(x) x is an
  injection from A to B.

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Sequences and Summations
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Sequences and Summations

• Theorem For any set A, A and P(A) have distinct
  cardinalities.
• Proof Assume that f A ? P(A). We shall show
  that f cannot be ONTO, that is there is a subset
  B ? A (an element of P(A)) such that ?b ? A, B ?
  f(b) (B does not have a preimage).
• For every a ? A, the subset f(a) either contains
  a or does not.
• Let B a ? A a ? f(a). B may be empty or
  not.
• If B f(b), for some b ? A, then
• if b ? B by the definition of B, b ? f(b) and if
  b ? f(b), b ? B. Since it is not possible for
  an element to belong and not belong to a set, we
  have a contradiction. So, b ? B.

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Sequences and Summations

• if b ? B by the definition of B, b ? f(b), which
  is also B. So b ? B. Since it is not possible
  for an element to belong and not belong to a set,
  we have here also a contradiction.
• The contradiction is a result of our assumption
  that B f(b). Hence B is not accounted for.
  QED
• Conclusion there are non-countable sets, for
  instance P(N).

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Sequences and Summations

• Examples
• (i) B Binary strings is not countable.
• (ii) F f f N ? 0,1 is not countable.
• (iii) 0,1 x x ? R ? 0 ? x ? 1 is not
  countable.

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Sequences and Summations

• The set of (finite length) strings S over a
  finite alphabet A is countably infinite.To show
  this we assume that
• A is nonvoid
• There is an alphabetical ordering of the
  symbols in A
• Proof List the strings in lexicographic order
• all the strings of zero length,
• then all the strings of length 1 in alphabetical
  order,
• then all the strings of length 2 in alphabetical
  order, etc.
• This implies a bijection from N to the list of
  strings and hence it is a countably infinite set.

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Sequences and Summations

• For example Let A a, b, c.
• Then the lexicographic ordering of A is l , a, b,
  c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab,
  aac, aba, .... f(0), f(1), f(2), f(3), f(4),
  . . . .

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Sequences and Summations

• The set of all C programs is countable .
• Proof Let S be the set of legitimate characters
  which can appear in a C program.- A C compiler
  will determine if an input program is a
  syntactically correct C program (the program
  doesn't have to do anything useful).- Use the
  lexicographic ordering of S and feed the strings
  into the compiler.
• If the compiler says YES, this is a
  syntactically correct C program, we add the
  program to the list.Else we move on to the next
  string.
• In this way we construct a list or an implied
  bijection from N to the set of C programs.
• Hence, the set of C programs is countable.
  Q. E. D.