\[ \newcommand{\C}{\mathbb{C}} \newcommand{\haar}{\mathsf{m}} \newcommand{\B}{\mathscr{B}} \newcommand{\D}{\mathscr{D}} \newcommand{\P}{\mathcal{P}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\g}{>} \newcommand{\l}{<} \newcommand{\intd}{\,\mathsf{d}} \newcommand{\Re}{\mathsf{Re}} \newcommand{\area}{\mathop{\mathsf{Area}}} \newcommand{\met}{\mathop{\mathsf{d}}} \newcommand{\emptyset}{\varnothing} \DeclareMathOperator{\borel}{\mathsf{Bor}} \DeclareMathOperator{\baire}{\mathsf{Baire}} \newcommand{\symdiff}{\mathop\triangle} \]

1. Prove that $\Lambda(\Q) = 0$.
2. Let $X$ be a non-empty set. Say that a subset $E$ of $X$ is big if $X \setminus E$ is countable, and small if $E$ is countable. Prove that \[ \{ A \subset X : A \textsf{ is small} \} \cup \{ A \subset X : A \textsf{ is big} \} \] is a σ-algebra on $X$.