I am a bio student self-studying Abbott's Understanding Analysis and would love some feedback on one of my answers to an exercise. I have no experience writing proofs, and I'm used to plug-n-chug math taught by school, but I'm determined to get through this book as I find it fascinating. Thanks!
Q: If $x ∈ (A ∩ B)^c$, explain why $x ∈ A^c ∪ B^c$. This shows that $(A ∩ B)^c ⊆ A^c ∪ B^c$
Pf: If $x \in A\cap B$, then $x \in A,B$ and $x \in A\cup B$. We can think of $A\cap B$ as the collection of elements in both $A$ and $B$. The complement $(A\cap B)^c$ is therefore the set of elements not in $A$ and $B$, elements can still originate from $A$ or $B$, just not those in both.
The set $(A\cap B)^c$ is equal to $A^c\cup B^c$ because $A^c$ is the set of elements not in A (but, again, can contain elements in $B$). But, if an element is in A and also in B, neither $A^c$ nor $B^c$ will contain that elements. Thus, $A^c\cup B^c$ can be thought of as the set of elements not in $A$ and $B$, just as with $(A ∩ B)^c$.