Notation for a size of a partitioning and mapping between set elements and partitions?

Question

Consider a infinite set of continuous numbers $A$. Now, I partition the set in N subsets $A_0, A_1, \cdots, A_N$. Thus, $A = \bigcup_{i=0}^N A_i$ and $\bigcap_{i=0}^N A_i = \emptyset$. We can say that $A = \{ A_0, A_1, \cdots, A_N\} $

I want to say 2 things about $A$, but I am not sure how to write mathematically.

1. I have a function $f: A \rightarrow \{0, 1, \cdots, N\}$ that maps $a \in A$ to a specified partition.
2. I want to describe $N$ by applying an operator to the partition e.g., $|A|$ is nice, but $|A|$ is not equal to $N$ but instead equal to the number of element of all partitions.

I am not sure how I can write these things.

Answer 1

There's a few issues with your setup (see my comment). Here I'm assuming $\mathcal{A} = \{A_0, A_1, \ldots, A_N\}$ is a partition of $A$ with the usual notation.

In that case, it would seem you are asking for

1. $f : A \to \{0, 1, \ldots, N\}$ is defined by $a \in A_{f(a)}$ for all $a \in A$
2. $N+1 = |\mathcal{A}|$

Note that the function in (1) is well-defined using the usual definition of "partition", but not well-defined as you've defined a "partition". As you've defined it, you would need instead to define it as something like $f(a) = \min\{i \mid a \in A_i, 0 \leq i \leq N\}$