The associative law states that for the logic formula:
$$(A \wedge B) \wedge C = A \wedge (B \wedge C)$$
$$(A \vee B) \vee C = A \vee (B \vee C)$$
I asked myself would the associative law hold for multiple operators, so I tested it out on $(A \wedge B) \vee C $ vs $A \wedge (B \vee C) $. This turned out to not be true once I did I truth table. For the first formula as long as C is true the entire thing is true even if A is not true, while for the second, A must be true in order for the logic formula to hold true.
My question is are there times when the associative law still works when I have multiple operators and not just two (e.g. $A \wedge B \wedge C \vee D\vee E$). How can I tell immediately whether or not the associative law is applicable? Are there operators (such as $\downarrow$) that work with the associative law even when I have different operators in the same formula?