$\{y : y \in \mathbb{N}$ and $y^2 < 100 < y\}$ is—

The condition requires $y^2 < 100 < y$, which means $y > 100$ while simultaneously $y^2 < 100$ (i.e. $y < 10$) — an impossible combination. Since no value satisfies both, the set has no elements and is empty.