Can I change or remove parentheses in a proposition consisting of conjunctions?

Question

Can I rewrite the first line as the second?

$$(A \land \lnot B) \land (C \land \lnot D)\tag{1}$$ and $$A \land \lnot B \land C \land \lnot D\tag{2}$$

Answer 1

Conjunction ("and", denoted "$\land$"), is indeed associative.

That means, you can move or remove parentheses among consecutive conjunctions within a proposition without changing the truth-value of that proposition. So yes,

$$(A \land \lnot B) \land (C \land \lnot D) \equiv A \land (\lnot B\land C) \land \lnot D \equiv A\land \lnot B \land C \land \lnot D$$

(Note that conjunction is also commutative, meaning $A\land B \equiv B\land A$)

In fact, see the Wikipedia entry addressing properties of logical connectives, for a summary.