Mathematics Notation Help: For each elements within a dataset, count all elements within and space time window

Question

So for every element in the current data set DC, has a space property i(s), and time property i(t).

For each of these elements, I count of elements with the space window (SW) and time window (TW).

How would I show this?

My attempt:

∀i ∈ DC = Ni(i)[ni|(i(s)∈ SW) ∩ (i(t) ∈ TW)]

Answer 1

So $DC$ is a set of elements. If each element has space propery, then you have a function $s: DC \to S$, where $S$ is a set of space values. So for every element $i ∈ DC$, it's space property is $s(i)$. Similarily with time. Your space window and time window are sets of some space/time values: $SW ⊆ S$, $TW ⊆ T$. All elements with the right space property forms a set $\{i ∈ DC: s(i) ∈ SW\}$, it's similary with time: $\{i ∈ DC: t(i) ∈ TW\}$. Their intersection $\{i ∈ DC: s(i) ∈ SW\} ∩ \{i ∈ DX: t(i) ∈ TW\} = \{i ∈ DC: s(i) ∈ SW ∧ t(i) ∈ TW\}$. If you want to count elements of some $X$, the notation $|X|$ may be used.