Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason - - PowerPoint PPT Presentation

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Countability Jason Filippou CMSC250 @ UMCP 06-23-2016 Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 1 / 12 Outline 1 Infinity 2 Countability of integers and rationals 3 Uncountability of R Jason Filippou (CMSC250 @ UMCP)

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Countability

Jason Filippou

CMSC250 @ UMCP

06-23-2016

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 1 / 12

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Outline

1 Infinity 2 Countability of integers and rationals 3 Uncountability of R

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 2 / 12

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Infinity

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 3 / 12

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Infinity

Infinite sets

Definition (Finite set) Let n ∈ N. A set A is called finite if and only if:

1 A = ∅, or 2 There exists a bijection from the set {1, 2, . . . , n} to A.

Definition (Infinite set) A set A is called infinite if, and only if, it is not finite.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 4 / 12

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Infinity

Countable sets

Definition (Countable set) Let A be any set. A is countable if, and only if:

1 A is finite, or 2 There exists a bijection from N∗ to A. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12

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Infinity

Countable sets

Definition (Countable set) Let A be any set. A is countable if, and only if:

1 A is finite, or 2 There exists a bijection from N∗ to A.

In the second case, A can also be called countably infinite.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12

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Infinity

Countable sets

Definition (Countable set) Let A be any set. A is countable if, and only if:

1 A is finite, or 2 There exists a bijection from N∗ to A.

In the second case, A can also be called countably infinite. Definition (Uncountable set) A set A is called uncountable if, and only if, it is not countable.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 5 / 12

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Countability of integers and rationals

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 6 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . .

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . .

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Let’s call this function f. We can make the following observations about f:

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Let’s call this function f. We can make the following observations about f:

1 No integer is counted twice! So, f is... ? Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Let’s call this function f. We can make the following observations about f:

1 No integer is counted twice! So, f is... ? 1-1. 2 All integers are (eventually) accounted for! So, f is... ? Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Let’s call this function f. We can make the following observations about f:

1 No integer is counted twice! So, f is... ? 1-1. 2 All integers are (eventually) accounted for! So, f is... ? onto. Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Z is countable

. . . −3 −2 −1 1 2 3 . . . . . . 7 5 3 1 2 4 6 . . . Let’s call this function f. We can make the following observations about f:

1 No integer is counted twice! So, f is... ? 1-1. 2 All integers are (eventually) accounted for! So, f is... ? onto.

From (1) and (2) we can deduce that the function is a bijection, and Z is countable.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 7 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . .

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . .

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . . Call this function g.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . . Call this function g. Is g onto?

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . . Call this function g. Is g onto? Is g 1-1?

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . . Call this function g. Is g onto? Is g 1-1? Therefore, g is a bijection from Z to Zeven.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Zeven is countable

. . . −6 −4 −2 2 4 6 . . . . . . −3 −2 −1 1 2 3 . . . Call this function g. Is g onto? Is g 1-1? Therefore, g is a bijection from Z to Zeven. So gof is a bijection from N to Zeven (formally prove at home)! Therefore, Zeven is countable.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 8 / 12

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Countability of integers and rationals

Is Q+ countable?

Reminder: Q+ = { m

n , m, ∈ N, n ∈ N∗}

Discuss it with your neighbors for a while!

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 9 / 12

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Uncountability of R

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 10 / 12

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Uncountability of R

Cantor’s diagonal argument

Famous proof by contradiction. Method known as diagonalization, or the diagonal argument.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 11 / 12

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Uncountability of R

The proof

Theorem R is uncountable. By contradiction.

Suppose that R is countable. This means that we can order all the reals in a list, as follows: 0.a11a12a13 . . . 0.a21a22a23 . . . 0.a31a32a33 . . . . . . Let us now create a real number r with decimal digits ri, which will be populated as follows: ri = 0, aii = 9 aii + 1, 0 ≤ aii < 9 By construction, r is different from all real numbers that we listed, since it’s guaranteed to be different from the i − th number at the i − th decimal digit, where i = 1, 2, . . . . Contradiction, because we assumed that we sequentially listed all the real numbers inside this very list. Therefore, R is uncountable.

Jason Filippou (CMSC250 @ UMCP) Countability 06-23-2016 12 / 12