EE1J2 - Slide 1 EE1J2 – Discrete Maths Lecture 10 Cardinality Uncountability of the real numbers

EE1J2 - Slide 2 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

EE1J2 - Slide 3 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!

EE1J2 - Slide 4 Uncountability of ℝ In fact there are uncountably many irrational numbers To see this we shall show that ℝ is uncountable. Then, since ℝ=ℚ  ℐ, ℐ must be uncountable In this case, | ℝ | > | ℚ |. The cardinality of ℝ is denoted by  1 – pronounced ‘aleph one’

EE1J2 - Slide 5 Cantor’s Proof The uncountability of ℝ was demonstrated by the mathematician George Cantor Cantor’s proof is a proof by contradiction We assume that ℝ is countable, and show that this leads to a contradiction

EE1J2 - Slide 6 Cantor’s 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

EE1J2 - Slide 7 Cantor’s 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

EE1J2 - Slide 8 Cantor’s 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) = x 1 f(2) = x 2 f(3) = x 3 …

EE1J2 - Slide 9 Cantor’s construction (contd) Write: kth digit in decimal expansion of x n

EE1J2 - Slide 10 Cantor’s 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)

EE1J2 - Slide 11 Cantor’s Construction (contd) Write: y = 0.(  d 1 1 )(  d 2 2 )(  d 3 3 )….(  d n n )…..

EE1J2 - Slide 12 Cantor’s 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, ‘9’s should not be chosen in the expansion to avoid rounding

EE1J2 - Slide 13 Summary of Lecture 10 Cardinality Uncountability Cantor’s proof of the uncountability of ℝ Definition of  1