# Countability

## Bijections

A **bijection **is a mapping between two sets such that there **exists **a **unique** pairing from a particular element of one set to another.

These ideas of **existence** and **uniqueness** can be formalized by considering some different types of maps:

A **bijection, **also known as an **isomorphism, **is a mapping that is **both one-to-one AND onto. **This guarantees that **the two sets must be the same size**, a statement known as the **isomorphism principle**

## Countability

#### Proving that something is countable:

Find a bijection from S to N or N to S (must prove one to one and onto). Note that a bijection in either direction is individually valid.

Find an injection from S to N, AND an injection from N to S.

**Enumeration**: list all elements of S.

## Enumerability

Let's now think about some ways we can list every element in a set. Some properties to keep in mind:

**Listing a set implies that it is countable.**Every element must have a unique, finite position on the list. (You can number them off.)

Any infinite set that can be listed is as large as the set of natural numbers.

One method of enumerating is to find a **recursive definition **of the set: that is, given any one element in the set, we can define the element that immediately follows it.

One example of a non-enumerable set is the set of all rational numbers: we can't write fractions in an order such that you can get to the next fraction in a finite number of steps. However, the *are* countably infinite (read on to find out why!)

## Pairs of Natural Numbers

### Rational Numbers

## Cantor's Diagonalization Argument

**Proof by contradiction:** assume that a set S is countable (even if it isn't). Then, there must exist a listing that contains all elements in the list (enumeration).

We can construct an item that isn't in the set by taking the diagonals of digits in the set:

This can be used to prove that real numbers are not countable.

**Formal steps :**

Assume that a set S can be enumerated.

Consider an arbitrary list of all the elements of S.

Use the diagonal from the list to construct a new element t.

Show that t is not in the list, but that t is in S.

This is a contradiction.

Here's a demonstration of the diagonalization argument in action:

**Continuum hypothesis:** there is no set with cardinality between the naturals and the reals.

**Generalized continuum hypothesis: **there is no infinite set whose cardinality is between the cardinality of an infinite set and its power set. (In other words, the power set of the natural numbers is not countable.)

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