How can I prove the generalized De Morgan laws using mathematical writing, and not induction (I am not allowed).
This is what I concluded:
$\bigcap_{i\in I}A_i' = (\bigcup_{i\in I}(A_i)).$
and its opposite:
$\bigcup_{i\in I}A_i' = (\bigcap_{i\in I}(A_i)).$
I know how to prove it using induction, but I am not allowed unfortunately. I am requested to prove only using logic (mathematical writing).
This is how I've generalized it: $\bigcup_{i\in I}A_i iff \exists \:i\left(i\in \:I\:\:∧\:x\in \:A_{_{_i}}\right)$
and the other one: $(\bigcap_{i\in I}(A_i)) iff \forall i(i\in I\:->\:X\in A_i)$
but I don't know how to prove it using logic for the generalized form.
Please show me how to do it correctly, I've written what I've already done.
Best Answer
We have that
\begin{align}x\in \bigcap_{i\in I}A_i'&\iff x\in A_i',\forall i\in I \\ &\iff x\notin A_i,\forall i\in I\\&\iff x\notin \bigcup_{i\in I} A_i\\&\iff x\in \left(\bigcup_{i\in I} A_i\right)' \end{align}
which shows $$\bigcap_{i\in I}A_i=\left(\bigcup_{i\in I} A_i\right)'.$$ Proceed in the same way to get the other equality.