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Exercise (Union And Intersection Properties)

Prove the statements in the Union and Intersection Properties theorem.

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  1. To prove $A\subseteq A$, let $a\in A$. Then clearly $a\in A$ which means $A\subseteq A$ as desired.
  2. We now prove $A\subseteq A\cup B$. Let $a\in A$. Then clearly $a\in A$ or $a\in B$. This means that $a\in A\cup B$, which proves that $A\subseteq A\cup B$.
  3. We now prove $A\cap B\subseteq A$. Let $x\in A\cap B$. This means $x\in A$ and $x\in B$. In particular, notice that we know $x\in A$. This completes the proof that $A\cap B\subseteq A$.
  4. Let $x$ be a real number. Let $P$ be the statement $x\in A$, and let $Q$ be the statement $x\in B$. Note that the statements $P\vee Q$ and $Q\vee P$ are logically equivalent from a truth table. The rule $A\cup B=B\cup A$ immediately follows. Alternately, we can prove this fact using the standard technique. Let $y\in A\cup B$. Then we know $y\in A$ or $y\in B$. This is equivalent to $y\in B$ or $y\in A$, which means $y\in B\cup A$. Hence we've shown $A\cup B\subseteq B\cup A$. The proof that $B\cup A\subseteq A\cup B$ is similar. Thus $A\cup B= B\cup A$.
  5. The proof that $A\cap B=B\cap A$ is almost identical to the previous.
  6. This is the next problem.
  7. This is the next problem.
  8. We now prove that $A\cup(B\cap C) = (A\cup B)\cap (A\cup C)$. We first show $A\cup(B\cap C) \subseteq (A\cup B)\cap (A\cup C)$. Let $x\in A\cup (B\cap C)$. This means that $x\in A$ or $x\in B\cap C$. We must show that $x\in A\cup B$ and that $x\in A\cup C$. There are two cases, namely $x\in A$ or $x\notin A$. Suppose first that $x\in A$. Then clearly $x\in A\cup B$ and $x\in A\cup C$ as $x$ is a member of the first set in each union. This shows that $x\in (A\cup B)\cap(A\cup C)$. Now suppose $x\notin A$. This means $x\in B\cap C$ (see the third sentences in this proof). Hence we know that $x\in B$ and $x\in C$. Since $x\in B$, we now $x\in A\cup B$. Since $x\in C$, we know $x\in A\cup C$. This shows that $x\in (A\cup B)\cap(A\cup C)$ as desired. This completes the proof that $A\cup(B\cap C) \subseteq (A\cup B)\cap (A\cup C)$. To finish, we must prove $(A\cup B)\cap (A\cup C) \subseteq A\cup(B\cap C)$. Let $y\in (A\cup B)\cap (A\cup C)$. We again use two cases. Suppose $y\in A$. Then clearly $y\in A\cup (B\cap C)$ by definition of union. The only other option is $y\notin A$. Recall we assumed that $y\in (A\cup B)\cap (A\cup C)$, which means $y\in (A\cup B)$ and $y\in (A\cup C)$. This means $y\in A$ or $y\in B$, and it means $y\in A$ or $y\in C$. Since we have assumed that $y\notin A$, this means that $y\in B$, and it means that $y\in C$. Together, this gives $y\in B\cap C$, which shows that $y\in A\cup (B\cap C)$.
  9. Your proof should be very similar to the previous.