ACIT4330-Page/content/Lectures/Lecture 3.md

lecture, date

lecture	date
3	2025-01-13

Proposition 1.1.4

Each real number is a limit of a sequence of rational numbers.

\mathbb{Q} \subset \mathbb{R}

As an ordered Number Field \mathbb{R} is complete, meaning that every Cauchy Sequence in \mathbb{R} converges to a real number. Equivalently, the real numbers have the Least Upper Bound Property; \forall X \subset \mathbb{R} bounded above has a least upper bound denoted by sup(X) \in \mathbb{R}. Eq. an inf(Y) \in \mathbb{R} if Y bounded below.

Functions and Cardinality

A function f : X \to Y is Injective if f(x) = f(y) \implies x = y.

Surjective if f(x) = y.

Bijective if it is both injective and surjective.

Then we write X \simeq Y.

|X| = |Y| \; \text{(cardinality)}

Say X is Countable if |X| = |\mathbb{N}|; this means that the members of X can be listed as a sequence with x_{n} = f(n), where f: \text{IN} \to X is some bijection.

Cantor's Diagonal Argument

The real numbers cannot be listed, or they are uncountable.

Indeed, present a list of the real numbers in \langle 0, 1 \rangle written as binary expansions. Then the number that has as its $n$-th digit, the opposite value to the $n$-th digit of the $n$-th number of the list, will never be in the list.

\
0	1	0	1	1	0	0	1	1	1	...
0	1	1	1	1	1	1	1	1	1	...
0	0	0	0	0	1	0	0	0	0	...
0	1	0	1	0	1	0	0	0	0	...
...
So here the bold numbers going diagonally from the \ shows that they cannot be countable as they are not the same number.
Cantor: 0.0011...

Axiom of Choice

Any Direct Product

\Pi_{i \in I} \, X_{i} \equiv \{ x : I \to \cup_{i \in I} X_{i} | x_{i} \equiv x(i) \in X_{i} \}

is non-empty when all x \neq \emptyset.

Any x \in \Pi_{i \in I} \, X_{i} is called a choice function.

The Power Set \wp(X) of X consists of all the subsets of X.

\exists bijection \wp(X) \to \Pi_{x} \{ 0, \, 1 \} = \{ f : X \to \{ 0, \, 1 \} \} that sends Y \subset X to its Characteristic Function.