# Converting This Into Set Notation

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.

• Writing $1 \leq...$ is redundant since all natural numbers are $\geq 1$. May 27, 2020 at 1:34
• "More compact" does not necessarily mean that it will be clearer to the reader. The original is definitely better than the new.
– Ted
May 27, 2020 at 2:23

...$$i,j,t \in \mathbb{N}_{\leq \max(I,J,T)}$$ would suffice.