# Boolean Algebra Negate Associative Law?

Given $$(A \vee B) \vee C = A \vee B \vee C = A \vee (B \vee C)$$ is true because of the associative law of linear algebra. I have a question about this rule in the case of a negative.

I have: $$\neg(A \vee (B \wedge C)) \vee (B \wedge C)$$

Treating the $$(B \wedge C)$$ as a single term, am I correct to apply the associative rule in this case resulting in $$\neg A \vee ((B \wedge C) \vee (B \wedge C))$$?

Checking my intial expression with the resulting one with an online simplifier tells me they are both equal, but I am not sure why that is.

• Later includes tautology no ? – user645636 Jan 21 '20 at 18:58

You cannot apply the associative property here. The negator encloses the whole of $$(A \vee (B\land C))$$ so you do not actually have an expression of the form $$X\vee Y \vee Z$$. Coincidentally, you did get the correct result but it would be derived as follows:
\begin{align*} &\neg(A \vee (B \land C)) \vee (B \land C)\\ =&(\neg A \land \neg(B\land C)) \vee (B\land C) \\ =& (\neg A \vee (B\land C)) \land (\neg (B\land C) \vee (B\land C))\\ =& \neg A \vee (B\land C). \end{align*}
using the distributive properties for "$$\neg$$", as well as for "$$\vee$$ and $$\land$$".