If i want to Prove $A^c \cup B^c$ = $(A \cap B)^c$ by a string of equalities
=$\{x|x\in A^c \cup B^c\}$
=$\{x|x\in A^c orx\in B^c\}$
=$\{x|x\notin A orx\notin B\}$
=$\{x|x\notin (A \cap B)\}$
=$\{x|x\in (A \cap B)^c\}$
1/ Is this proof ok as it stands?
2/ if i wanted to justify line 4 would suppose $x\in (A \cap B)$ then $x\in A$ and $x\in B$ which contradicts line 3 be ok.
3/ as this i s a string of equalities would it be ok to put that argument beside line 4? Thanks
I would add a line in between to better show the logic ... and that will also make your question about the justification more clear:
... =$\{x|x\notin A \text{ or } x\notin B\}$
$\color{red}{=\{x|\text{ it is not the case that } x\in A \text{ and } x\in B\}}$
=$\{x|x\notin (A \cap B)\}$
=$\{x|x\in (A \cap B)^c\}$
The logical justification of going from
$\{x|x\notin A \text{ or } x\notin B\}$
to
$\{x|\text{ it is not the case that } x\in A \text{ and } x\in B\}$
is by logical DeMorgan: in abstract terms, you have a statement of the form $\neg p \lor \neg q$, which is equivalent to $\neg (p \land q)$