Do De Morgan's laws hold for arbitrary infinite expressions with union and intersection?

It can be shown that De Morgan’s laws hold for infinite union and infinite intersection:

$$ left( bigcup_{i in I} A_i right)^c = bigcap_{i in I} A_i^c tag{1} $$
$$ left( bigcap_{i in I} A_i right)^c = bigcup_{i in I} A_i^c tag{2} $$

even if the index set $I$ is uncountable. Here superscript $c$ denotes complement w.r.t. universe $U$.

Now consider an arbitrary expression of the form

$$ A_1 cup A_2 cap A_3 cup A_4 cup … $$

I will denote union with $0$ and intersection with $1$ so the form of above expression looks like $0100…$

Some such expressions can be converted to laws $(1), (2)$, for example, complement of $101010101…$ simply means

$$ left( bigcup_{i in mathbb{N}} (A_{2i-1} cap A_{2i}) right)^c = bigcap_{i in mathbb{N}} (A_{2i-1} cap A_{2i})^c = bigcap_{i in mathbb{N}} (A_{2i-1}^c cup A_{2i}^c) $$

But what about other examples like $011011100101110111…$ (the string contains all binary numbers in order)? Can we rewrite complements of all such expressions in terms of $A_i^c$?

Edit: Assume that $cap$ takes precedence over $cup$, i.e., $A cap B cup C = (A cap B) cup C$