On the complement of a union of cosets

Let $g_1{H}_1,\dots ,g_n{H}_n$ be left cosets of subgroups of a group $G$ whose union $U$ is a proper subset of $G$. We prove that $G$ is the union of at most $2^n$ left translates of $A:=G\setminus U$. For finite $G$ this yields $\mathrm{\mid }A\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }\mathrm{/}{2}^n$, which settles a conjecture of Tărnăuceanu and the author. Equality $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}{2}^n$ can only hold when $A$ is a left coset of $K:=H_1\cap \dots \cap H_n$. Moreover, every subgroup occurring in an irredundant cover of an arbitrary group by $n$ cosets has index at most $2^{n-1}$. This improves a lemma of Neumann.