Set Numbers
Why
We want to count the number of elements in a set.
Defining result
A set can be equivalent to at most one
natural number.1
The number (or size) of a finite set is the unique natural number equivalent to it.
Notation
We denote the number of a set by $\num{A}$. Equally good notation, which we will not use in these sheets, is $\#(A)$.
Restriction to a finite set
If we restrict $E \mapsto \num{E}$ to the domain $\powerset{X}$ of some set $X$ then $\num{\cdot }: \powerset{X} \to \omega $ is a function.2
Properties
$A \subset B \implies \num{A} \leq \num{B}$