EE1J2 Discrete Maths Lecture 10

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EE1J2 Discrete Maths Lecture 10

• Cardinality
• Uncountability of the real numbers

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Uncountable Sets

• Recall that a set A is countable if and only if
  there exists a bijection
• f N?A
• Equivalently, A is countable if it can be written
  as a list
• If A is not countable (i.e. no bijection f N?A
  exists) then A is called uncountable

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The Real Numbers R

• The set of real numbers R contains the set of
  rational numbers as a subset Q ? R
• Real numbers which are not rational are called
  irrational. Let I R- Q
• Examples
• Intuitively you might think that there are not
  many of them you would be wrong!

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Uncountability of R

• In fact there are uncountably many irrational
  numbers
• To see this we shall show that R is uncountable.
  Then, since RQ? I, I must be uncountable
• In this case, R gt Q. The cardinality of R is
  denoted by ?1 pronounced aleph one

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Cantors Proof

• The uncountability of R was demonstrated by the
  mathematician George Cantor
• Cantors proof is a proof by contradiction
• We assume that R is countable, and show that this
  leads to a contradiction

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Cantors Proof

• So, suppose that R is countable
• Then there is a bijection f N ? R
• In particular, f is onto (surjective), so for
  every real number x ? R, there exists n ? N such
  that f(n) x

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Cantors Proof (cont.)

• Cantor showed how to construct a real number y
  such that there is no n ? N such that f(n) y
• This will contradict the fact that f is
  surjective
• So, assuming that R is countable leads to a
  contradiction
• Therefore R must be uncountable

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Cantors construction

• Suppose R is countable.
• Then 0,1 is certainly countable
• Let f N ? 0,1 be a bijection
• Use f to write 0,1 as a list
• f(1) x1
• f(2) x2
• f(3) x3

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Cantors construction (contd)

• Write

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Cantors construction (contd)

• The number y is constructed as follows
• Consider the decimal expansion of y
• For each n, choose the number in the nth position
  in the decimal expansion of y to be different to
  the number in the nth position in the decimal
  expansion of f(n)

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Cantors Construction (contd)

• Write

y 0.(?d11)(?d22)(?d33).(?dnn)..
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Cantors Proof (cont.)

• By construction, y differs from f(n) in at least
  the nth decimal place for every n
• This contradicts the assertion that f maps N onto
  R (i.e that f is a surjection)
• Note
• Some care is needed in the construction of y
• For example, 9s should not be chosen in the
  expansion to avoid rounding

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Summary of Lecture 10

• Cardinality
• Uncountability
• Cantors proof of the uncountability of R
• Definition of ?1