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\section*{Week 1}

\exercise*[0.3.6]
\begin{enumerate}[label=\emph{\alph*)}]
    \item Wanting to show:
        $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

        In order to prove this equivalence, we have to prove the implication both ways. We use two lemmas for this.

        \begin{lemma}
            \label{lem:set1}
            $A \cap (B \cup C) \implies (A \cap B) \cup (A \cap C)$
        \end{lemma}

        Let $x \in A \cap (B \cup C)$.
        By the definition of set intersection, $x \in A$ and $x \in B \cup C$.
        By the definition of set union, $x \in A$ and $(x \in B$ or $x \in C)$.
        From propositional logic we know that for propositions $P$, $Q$ and $R$ the following holds:
        $P \land (Q \lor R) \iff (P \land Q) \lor (P \land R)$.
        So, substituting for this particular case yields $(x \in A$ and $x \in B)$ or $(x \in A$ and $x \in C)$.
        Using the definition of set intersection again gets $x \in A \cap B$ or $x \in A \cap C$. 
        Using the definition of set union again gives $x \in (A \cap B) \cup (A \cap C)$. \qed

        \begin{lemma}
            \label{lem:set2}
            $(A \cap B) \cup (A \cap C) \implies A \cap (B \cup C)$
        \end{lemma}
        Let $x \in (A \cap B) \cup (A \cap C)$.
        By the definition of set union, $x \in (A \cap B)$ or $x \in (A \cap C)$.
        By the definition of set intersection, $(x \in A$ or $x \in B)$ and $(x \in A$ or $x \in B)$.
        Using the same propositional logical equivalence as in Lemma \ref{lem:set1}, this gives 
        $x \in A$ and $(x \in B$ or $x \in C)$.
        Wrapping up, we use the definition of set union to get $x \in A$ and $x \in B \cup C$ and
        the definition of intersection to get $x \in A \cap (B \cup C)$. \qed

        Using Lemma \ref{lem:set1} and \ref{lem:set2}, we get the desired equivalence of 
        $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. \qed
    \item Wanting to show:
        $A \cup (B \cap C) = (A \cup B) \cap (a \cup C)$
        
        This proof is so similar to a) that it feels like a waste of time and will therefore be left to the reader.
\end{enumerate}

\hrule

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