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Consider the coordinate grid of points between $(-5, -5)$ and $(5, 5)$, inclusive, spaced 0.3 units apart in both directions. Is there a more compact way of notating this without writing something long, such as $\{(-5, -5), (-5, -4.7), \dots, (-5, 5), (-4.7, -5), \dots, (5, 5)\}$?

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You could use $(0.3\mathbb{Z}-5\cap[-5,5])^2$

Explanation: $\mathbb{Z}$ are the integers, and $0.3\mathbb{Z}$ are just values which are $0.3$ units apart. The $-5$ just moves the origin of the lattice around. Intersecting the lattice with $[-5,5]$ restricts it to the intended bounds. Finally, you take the Cartesian product with itself, aka square it, to get two dimensional.

PS: not both, 5 and -5 can be part of the solution, since $10/0.3=33$ remainder $0.1$.

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