Cardinality 2/12 Example: Do the sets Do the sets Cardinality { - - PowerPoint PPT Presentation

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Cardinality

Lecture 14 (Chapter 19) October 26, 2016

2/12 / department of mathematics and computer science

Cardinality

Example:

Do the sets {5, 10, 15, . . . , 155} {10, 20, 30, . . . , 310} {1, 2, 3, . . . , 31} {0, 1, 2, . . . , 30} have the same size? Do the sets N and Z have the same size? Two sets X and Y have the same cardinality (=have the same size) if there exists a bijection from X to Y . X ∼ Y means ’X and Y have the same cardinality’

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Countable

A set is called countable if it has the same cardinality as N. Is Z countable?

1 2 3 . . . ↓ ↓ ↓ ↓ −3 −2 −1 1 2 3 . . . no bijection 1 2 3 4 5 6 . . . ↓ ↓ ↓ ↓ ↓ ↓ ↓ −1 1 −2 2 −3 3 . . . bijection

Yes, via the bijection f : N → Z defined by f(n) = 1

2n

if n is even − 1

2(n + 1)

if n is odd

5/12 / department of mathematics and computer science

Countable

A set is called countable if it has the same cardinality as N. Is N2 countable?

1 2 3 . . . ↓ ↓ ↓ ↓ (0, 0) (0, 1) (0, 2) (0, 3) . . . no bijection (0, 0) (0, 1) (0, 2) (0, 3) . . . (1, 0) (1, 1) (1, 2) (1, 3) . . . (2, 0) (2, 1) (2, 2) (2, 3) . . . (3, 0) (3, 1) (3, 2) (3, 3) . . . . . . . . . . . . . . . ... 2 5 9 . . . 1 4 8 13 . . . 3 7 12 18 . . . 6 11 17 24 . . . . . . . . . . . . . . . ...

Yes, via the bijection f : N2 → N defined by f(x, y) = x+y

i=0 i

• + y = 1

2(x + y)(x + y + 1) + y

6/12 / department of mathematics and computer science

Countable

A set is called countable if it has the same cardinality as N. Is Q+ countable?

1 1 1 2 1 3 1 4

. . .

2 1

X

2 2 2 3

X

2 4

. . .

3 1 3 2

X

3 3 3 4

. . .

4 1

X

4 2 4 3

X

4 4

. . . . . . . . . . . . . . . ... The bijection from N to Q+ is similar to the bijection from N to N2, except that n

d has to be skipped if n and d are not relatively prime.

9/12 / department of mathematics and computer science

Infinite 0-1-sequences

Theorem:

The set of infinite 0-1-sequences is not countable.

Proof:

Reductio ad absurdum: we suppose that the set of infinite 0-1-sequences is countable, and derive a contradiction. If the set of infinite 0-1-sequences is countable, then there exists an enumeration, i.e., a bijection from N to that set:

1 2 3 . . . ← N ↓ ↓ ↓ ↓ ϕ0 ϕ1 ϕ2 ϕ3 . . . ← all 0-1-sequences without repetition

[Proof continues on next slide.]

10/12 / department of mathematics and computer science

Infinite 0-1-sequence (continued proof)

List those sequences ϕ0, ϕ1, ϕ2, ϕ3, . . . :

ϕ0 = a00 a01 a02 a03 · · · ϕ1 = a10 a11 a12 a13 · · · ϕ2 = a20 a21 a22 a23 · · · ϕ3 = a30 a31 a32 a33 · · · . . . . . .              every aij is either 0 or 1

Now consider the diagonal sequence

ψ = a00 a11 a22 a33 · · ·

and in it replace every 0 by a 1 and vice versa:

χ = b0 b1 b2 b3 · · · met bi = aii .

Then no ϕi can be equal to χ, so χ is not in the enumeration. But χ is an infinite 0-1-sequence; so it must be in the enumeration. Contradiction!

11/12 / department of mathematics and computer science

Uncountable

An infinite set that is not countable, is called uncountable. Examples: {0, 1}N, NN, 0, 1, R, P(N), . . . Degrees of infinity: ℵ0 : N, Z, Q, N2, Z2, Q × Z, . . . countable ℵ1 : R, P(N), {0, 1}N, NN, . . . uncountable ℵ2 : P(R), P(P(N)), . . . uncountable NB: N ∼ R ∼ P(R).

12/12 / department of mathematics and computer science

Exercises

Exercise 1

Prove that R+ has the same cardinality as R.

Exercise 2

Prove that every equivalence class associated with the equivalence relation ≡6 is countable.

Exercise 3

Prove that the set N × {0, 1} is countable.

Exercise 4

Prove that if V and W are countable, then also V × W is countable.

Exercise 5

Prove that the set NN of all functions from N to N is uncountable.