Simpliest Propositional Equivalences proof question

Question

I'm solving some propositional equivalences questions and I just want to make sure that following two logics.

If,

• $p \land q = q \land p$

• $p \vee q = q \vee p$

  • in any case, are correct because of the commutativity rule,

Than,

• $\neg p \vee \neg q = \neg q \vee \neg p$
• $\neg p \wedge \neg q = \neg q \wedge \neg p$

are correct as well?

I got most of the questions right by using them but I'm double checking it was not a luck.

It is so sad I don't know how to make a mathematical symbols since I'm pretty new here.

Answer 1

Yes, that's correct. The commutativity of $\land$ and $\lor$ tells us that for any $a, b$ we have that: \begin{align*} a \land b &= b \land a \\ a \lor b &= b \lor a \end{align*} In particular, taking $a = \neg p$ and $b = \neg q$ yields: \begin{align*} \neg p \land \neg q &= \neg q \land \neg p \\ \neg p \lor \neg q &= \neg q \lor \neg p \end{align*}