How to write this sentence formally

Question

I have this sentence

• Prediction is a numerical value, $P_{a,j}$, expressing the predicted likeliness of item $i_{j} \notin I_{u_{a}}$ for the active user $u_a$. This predicted value is within the same scale (e.g., from $1$ to $5)$ as the opinion values provided by $u_{a}$

I want to write the Pa,i (the prediction of a user a on an item i) formally.

because P is a rating, thus P belongs to the group ratings.

Rating = {0,1,2,3,4,5}

P belong-to-group-symbol Rating

I also know that a belongs to Users. Thus.

U = {u1, u2, .... um}
a belong-to-group-sybmo U

and I know that i belongs to Items. Thus,

I = {i1, i2, .... in}
i belong-to-group-symbol I

my question

is how can I say that P belongs to Rating, and at the same time say that P is a relationship between a (user) and i (item)

Answer 1

Let $R$ be the space of ratings, $U$ the space of users and $I$ the space of items. Then, $P:U \times I \to R$ is a function that takes elements $(a,i) \in U \times I$ into an element of $R$.