Fundamentals of Pure Mathematics - University of St …alanc/teaching/mt3600/mt3600... · Prefatory remarks Fundamentals of Pure Mathematics These slides can be downloaded from: ˜alanc/teaching/default.html

Prefatory remarks

Fundamentals of Pure Mathematics

These slides can be downloaded from:

www-groups.mcs.st-andrews.ac.uk/˜alanc/teaching/default.html

Alan J. Cain (Rm. 331) Fundamentals of Pure Mathematics 2006 1 / 29

Cardinality and countability

Questions

Let A = {1, 2, 3} and B = {4, 5, 6}.

Are A and B the same size?

(This is not a trick question.)

Yes. They are the same size: both A and B have three elements.



Questions

Let A = {1, 2, 3} and C = {7, 8}.

Are A and C the same size?

(No, this is not a trick question either.)

No. C is smaller: it contains two elements and A contains threeelements.



Sizes of finite sets

When do two sets have the same size? When is one set larger thananother?

For finite sets, this is easy: X and Y have the same size if they containthe same number of elements. X is larger than Y if X contains agreater number of elements.

Example

Let A = {1, 2, 3}, B = {4, 5, 6}, C = {7, 8}. Then A is the same size asB and larger than C.



Important questions

Why is this so boring?

Because we’re just laying a foundation for something moreinteresting. . .

What about infinite sets?

What can we say about the relative ‘sizes’ of N and2N = {2k : k ∈ N}?



Comparing sizes without counting

Example

Let A = {1, 2, 3}, B = {4, 5, 6}, C = {7, 8}.

We can find a bijection between A and B: defined f : A → B by:

f (1) = 4, f (2) = 5, f (3) = 6.

This ‘pairs off’ elements of A and B:

A 1 2 3l l l

B 4 5 6

So A and B have the same size.



Comparing sizes without counting

Example

Let A = {1, 2, 3}, B = {4, 5, 6}, C = {7, 8}.

We can find an injection from C to A: define g : C → A by:

g(7) = 1, g(8) = 2.

This ‘pairs off’ elements of C and A:

C 7 8l l

A 1 2 3

But 3 is left over. So C is smaller than A.



Comparing sizes without counting – infinite sets

What can we say about the relative ‘sizes’ of N and 2N = {2n : n ∈ N}?

We can define a bijection f : 2N → N with f (n) = n/2. This pairs offelements of N and 2N:

2N 2 4 6 8 10 12 . . .l l l l l l

N 1 2 3 4 5 6 . . .

So 2N is ‘the same size’ as N.



Comparing sizes without counting – infinite sets

What can we say about the relative ‘sizes’ of N and 2N = {2n : n ∈ N}?

We can define an injection g : 2N → N by g(x) = x . This pairs offelements of 2N with elements of N

2N 2 4 6 . . .l l l l l l

N 1 2 3 4 5 6 . . .

This leaves 1, 3, 5, . . . ‘unpaired’. So 2N is ‘smaller’ than N.



History

Galileo concluded that it was impossible to compare infinitequantities and abandoned further thought on the subject. He says‘infinity and indivisibility are in their very nature incomprehensibleto us’.

Leibniz considered the same question and concluded that thenotion of the size of the set N is self-contradictory.



Cardinality

Definition 19.1

Sets A and B have the same cardinality if there is a bijection f : A → B.In this case, write |A| = |B|.

We will also say informally ‘A and B are the same size’ when |A| = |B|.

We have not defined |A|. (Too technical — beyond the scope of thiscourse.)



Examples

Example 19.2

Let A = {1, 2, 3}, B = {4, 5, 6}, C = {7, 8}. Then |A| = |B| since

f : A → B; 1 7→ 4, 2 7→ 5, 3 7→ 6

is a bijection.



Examples

Example 19.3

The map i : N → N, n 7→ n is a bijection. So |N| = |N|.

Example 19.4

The map f : 2N → N, n 7→ n/2 is a bijection. So |N| = |2N|.



Examples

Example 19.5

Let X = {5, 6, 7, . . .}. The map g : N → X , n 7→ n + 4 is a bijection. So|N| = |X |.

Example 19.6

Let Y = {k2 : k ∈ N}. The map h : N → Y , n 7→ n2 is a bijection. So|N| = |Y |.



Examples

Example 19.7

The map f : N → Z,

n 7→

{

n/2 if n is even,

−((n − 1)/2) if n is odd,

is a bijection. So |N| = |Z|.

Example 19.8

The map tan : (−π/2, π/2) → R is a bijection. So |(−π/2, π/2)| = |R|.



Countability

We will be particularly interested in sets that have the same cardinalityas N.

Definition 19.9

Let A be an infinite set. If |N| = |A|, then A is said to be countable, orcountably infinite.

We have seen that |N| = |N|, |N| = |2N|, |N| = |Z|, and|N| = |{k2 : k ∈ N}|.

So N, Z, 2N, {k2 : k ∈ N} are all countable.



Countability

Remark 19.10

For some authors, ‘A is countable’ means ‘A is finite or countablyinfinite’ (i.e. A is finite or |A| = |N|. For us, ‘A is countable’ alwaysmeans ‘A is countably infinite’.



Countability

Example 19.11

Let P = {2, 3, 5, 7, 11, 13, . . .} be the set of prime numbers.Let pn be the nth prime number. Then define f : P → N by pn 7→ n.f is a bijection. So |P| = |N|. Thus P is countable.



Countability

Why ‘countable’?

Because a bijection f : N → X gives us a way of ‘counting’ X .

We point to f (1) and count 1,



We point to f (3) and count 4,...

We point to f (n) and count n,...



Countability

‘Counting’ or ‘enumerating’ elements of Z:

1 → 0 8 → 4 15 → −7

2 → 1 9 → −4 16 → 8

3 → −1 10 → 5 17 → −8

4 → 2 11 → −5 . . .

5 → −2 12 → 6

6 → 3 13 → −6

7 → −3 14 → 7

It is acceptable to prove a set is countable by showing that one can‘count’ the elements like this.



Countability

Example 19.12

Let B be set of finite binary sequences:{b1b2 · · · bn : bi ∈ {0, 1}, n ∈ N ∪ {0}}

Enumerate the sequences as follows:

1 → 8 → 0, 0, 0 15 → 1, 1, 1

2 → 0 9 → 0, 0, 1 16 → 0, 0, 0, 0

3 → 1 10 → 0, 1, 0 17 → 0, 0, 0, 1

4 → 0, 0 11 → 0, 1, 1 . . .

5 → 0, 1 12 → 1, 0, 0

6 → 1, 0 13 → 1, 0, 1

7 → 1, 1 14 → 1, 1, 0

So B is countable.



Countability of N × N

Theorem 19.13N × N is countable.

1(1, 1) 2(1, 2) 4(1, 3) 7(1, 4) . . .ւ ւ ւ ւ

3(2, 1) 5(2, 2) 8(2, 3) 12(2, 4) . . .ւ ւ ւ

6(3, 1) 9(3, 2) 13(3, 3) 18(3, 4) . . .ւ ւ

10(4, 1) 14(4, 2) 19(4, 3) 25(4, 4) . . .... ւ

......

...



More cardinality

Recall that sets A and B have the same cardinality if there is a bijectionf : A → B. In this case, write |A| = |B|.

Definition 19.14

The cardinality of A is less than or equal to the cardinality of B if thereis an injection f : A → B. (Write |A| ≤ |B|.)

The cardinality of A is strictly less than the cardinality of B if there is aninjection f : A → B but there is no bijection from A to B. (Write|A| < |B|.)



Examples

Example 19.15

Let A = {1, 2, 3}, C = {7, 8}. Then |C| < |A| since

g : C → A; 7 7→ 1, 8 7→ 2

is an injection and it is clear that there is no bijection from C to A.



Examples

Example 19.16

Recall that i : 2N → N, i(n) = n is an injection. So |2N| ≤ |N|.In fact, we know that |N| = |2N| since there is a bijection betweenthese two sets.



Examples

Example 19.17

Recall that there are injective mappings

f : N → Z, g : Z → Q, h : Q → R, i : R → C.

So|N| ≤ |Z|, |Z| ≤ |Q|, |Q| ≤ |R|, |R| ≤ |C|.

Or, more compactly,

|N| ≤ = |Z| ≤ |Q| ≤ |R| ≤ |C|.



A basic result

Proposition 19.18

If X is a finite set, then |X | < |N|.

Proof.

Suppose X = {x1, x2, . . . , xn}.Define f : X → N by xi 7→ i . Then f is an injection. So |X | ≤ |N|.There is no bijection from X to N. So |X | < |N|.



Basic results

Theorem 19.191 If |A| = |B| and |B| = |C| then |A| = |C|;2 If |A| = |B| and |B| < |C| then |A| < |C|;3 If |A| < |B| and |B| = |C| then |A| < |C|;4 If |A| < |B| and |B| < |C| then |A| < |C|.

Proof.1 Let f : A → B and g : B → C be bijections. Then g ◦ f : A → C is a

bijection.2 Let f : A → B be a bijection and g : B → C be an injection. Then

g ◦ f : A → C is an injection. Suppose h : A → C is a bijection.Then h ◦ f−1 : B → C is a bijection, which is a contradiction.



Cardinality of N

Theorem 19.20

N has the smallest cardinality of any infinite set. That is, |N| ≤ |A| forany infinite set A.

Proof.

Let A be infinite. Define a map f : N → A inductively as follows:

Choose any a ∈ A and define f (1) = a.

Suppose f (1), . . . , f (n) have been defined.Choose b ∈ A − {f (1), . . . , f (n)} and define f (n + 1) = b. (We cando this because A is infinite.

The map f is injective. So |N| ≤ |A|.


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Fundamentals of Pure Mathematics - University of St …alanc/teaching/mt3600/mt3600... · Prefatory remarks Fundamentals of Pure Mathematics These slides can be downloaded from: ˜alanc/teaching/default.html