Exercise 1.2.5

Question

[De Morgan's Laws]
  \label{demorgan}
  Let $A$ and $B$ be subsets of $\mathbf R$.
  \begin{enumerate}[label=(\alph*)] 
  \item If $x \in(A \cap B)^{c}$, explain why $x \in A^{c} \cup B^{c}$. This shows that $(A \cap B)^{c} \subseteq$ $A^{c} \cup B^{c}$

\item Prove the reverse inclusion $(A \cap B)^{c} \supseteq A^{c} \cup B^{c}$, and conclude that $(A \cap B)^{c}=A^{c} \cup B^{c}$
  \item Show $(A \cup B)^{c}=A^{c} \cap B^{c}$ by demonstrating inclusion both ways.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item If $x \in (A \cap B)^c$ then $x 
otin A \cap B$ so $x 
otin A$ or $x 
otin B$ implying $x \in A^c$ or $x \in B^c$ which is the same as $x \in A^c \cup B^c$.
  \item Let $x \in A^c \cup B^c$ implying $x \in A^c$ or $x \in B^c$ meaning $x 
otin A$ or $x 
otin B$ implying $x 
otin A \cap B$ which is the same as $x \in (A \cap B)^c$.
  \item First let $x \in (A \cup B)^c$ implying $x 
otin A \cup B$ meaning $x 
otin A$ and $x 
otin B$ which is the same as $x \in A^c$ and $x \in B^c$ which is just $x \in A^c \cap B^c$. Second let $x \in A^c \cap B^c$ implying $x \in A^c$ and $x \in B^c$ implying $x 
otin A$ and $x 
otin B$ meaning $x 
otin A \cup B$ which is just $x \in (A\cup B)^c$.
   \end{enumerate}

Exercise 1.2.5

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Exercise 1.2.5

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