I have a vector a vector of length $n$and can only contain values $\{A,B,C\}$. I'm representing this mathematically as: $$\mathbf{x} \in \{A,B,C\}^{1\times n}$$ Question 1: Am I properly mathematically representing what I am saying in words?
I first map this vector to numerical values with the following mapping function: $f: \{A,B,C\} \rightarrow \{2,4,8\} $
Question 2: I've seen mapping functions written this way but is there a simpler, more intuitive way to do it?
Then, I take $f(\mathbf{x})$ and convert into a one-hot vector format such that: $$OneHot(f(x)) \in \mathbb{R}^{v\times n}$$ where $v$ is the size of the vocabulary of $\mathbf{x}$. In this case, $v=3$.
Question 3: How can I mathematically represent one hot vector notation? I do see some option in this post. Is there a more intuitive way to represent it? The representation in the post isn't immediately obvious for a non pure math person.