1. ## complement help

I'm confused about these problems. How would I prove?

$\displaystyle (A \cup B \cup C)^c = A^c \cap B^c \cap C^c$

and

The complement of the complement of A = A.

I drew a venn diagram, and I can see from the diagram that they are equal. I'm just not sure how to approach this from a proof standpoint.

2. For both problems, show containment in both directions. I'll help get you started.

$\displaystyle x\in (A\cup B\cup C)^c$

$\displaystyle \Rightarrow x\notin A\cup B\cup C$

$\displaystyle \Rightarrow x\notin A\ \text{and}\ x\notin B\ \text{and}\ x\notin C$

$\displaystyle \Rightarrow x\in A^c\ \text{and}\ x\in B^c\ \text{and}\ x\in C^c$

$\displaystyle \Rightarrow x\in A^c\cap B^c\cap C^c$

$\displaystyle \Rightarrow (A\cup B\cup C)^c\subset A^c\cap B^c\cap C^c$

3. Thanks. So, would the other side be...

$\displaystyle x \in A^c \cap B^c \cap C^c$

$\displaystyle \Leftarrow x \notin (A \cap B \cap C)$

$\displaystyle \Leftarrow x \notin A$ and $\displaystyle x \notin B$ and $\displaystyle x \notin C$

$\displaystyle \Leftarrow x \in A^c$ and $\displaystyle x \in B^c$ and $\displaystyle x \in C^c$

$\displaystyle \Leftarrow x \in A^c \cup B^c \cup C^c$

$\displaystyle \Leftarrow x \in (A \cup B \cup C)^c$

4. Not quite.

Remove your second statement, $\displaystyle x\notin (A\cap B\cap C)$, as the third follows directly from the first. If you'd like to be precise, maybe replace $\displaystyle x\notin (A\cap B\cap C)$ with $\displaystyle x\in A^c\ \text{and}\ x\in B^c\ \text{and}\ x\in C^c$, your fourth statement.
Think about $\displaystyle x\notin A\ \text{and}\ x\notin B\ \text{and}\ x\notin C$. Since $\displaystyle x$ isn't in a single of the three sets, then it cannot be in their union. So your fourth statement should be $\displaystyle x\notin (A\cup B\cup C)$. This then implies $\displaystyle x$ is on the set you want.

I highly suggest writing out a sentence or two justification for each step; because x is in this set..., by this definition..., by this theorem..., ect. These ideas will keep reoccurring throughout math, but there will not always symbolic logical connections from each step to the next, it your job to convince not only the reader of your proof, but more importantly yourself, that what you have written down is right.

Hope this helps.

Have you done your second problem, $\displaystyle (A^c)^c=A$ ?