I have the following sentence that I would like to make more compact by using set notation. I'm a bit rusty on using set notation though, so i'm not quite sure how to formulate it in a compact manner. For context, I am trying to describe a situation in discrete space on a lattice grid and in discrete time.
original text
"...total size of the grid is $I\times J$ where $i = 1,...,I$ and $ j = 1, ... ,J$ index rows and columns, respectively. Variables that change over time are sub-scripted by $t$ where $t = 1, ... , T$."
new text
"...total size of the grid is $I\times J$ where $\{ i,j \in \mathbb{N}\thinspace |\thinspace 1\le i\le I, 1\le j \le J \}$ where $i$ and $j$ index rows and columns, respectively. Variables that change over time are sub-scripted by $t$ where $\{ t \in \mathbb{N}\thinspace |\thinspace 1\le t \le T\}$."
I'm just hoping to see if maybe there is a better way I can write the definitions of i,j and t in a more compact way then what I have right now. I'd be open to any suggestions on how I can use some notational artistry to improve the sentence.
...$i,j,t \in \mathbb{N}_{\leq \max(I,J,T)}$ would suffice.