We prove that the perturbation class of the upper semi-Fredholm operators from $X$ into $Y$ is the class of the strictly singular operators, whenever $X$ is separable and $Y$ contains a complemented copy of $C[0, 1]$. We also prove that the perturbation class of the lower semi-Fredholm operators from $X$ into $Y$ is the class of the strictly cosingular operators, whenever $X$ contains a complemented copy of $\ell_1$ and $Y$ is separable. We can remove the separability requirements by taking suitable spaces instead of $C[0, 1]$ or $\ell_1$.