Is this the correct way to represent this set using this set builder notation?
$$\big\{1, 4, 9, 16, ..., 100\} \cap \big\{1, 3, 5, 7, ...\}$$
$$A=\big\{2x - 1 \wedge y^2 \; | \; x \in \mathbb{N}, y \in \mathbb{N}, x \leq 10\}$$
I am confused as to how to set one variable to describe an infinite range and another to describe a finite one.
The $\wedge$ operator is from logic and applies to propositions or boolean variables, but doesn't apply to numbers.
Either you use the existential quantifier, as in $$\{n\in \mathbb{N}\ |\ (\exists k\in\mathbb{N},(n=k^2)\wedge(k\leq10))\wedge (\exists \ell\in\mathbb{N},n=2\ell-1)\}$$
Or you can use the fact that the square of an integer is odd if and only if that integer is odd:
$$\{(2n-1)^2\ |\ n\in\mathbb{N},n\leq5\}$$