Simplfiy $[[[(p\land q)\land r)]\lor [(p\land q)\land\lnot r)]]\lor\lnot q]\to s$
I know how to do it, but am stuck on the last few simplifications.
Below are a few things I am a bit confused on/ unsure if I'm correct
For the first step, do I ignore the "or not q" and simplfiy everything on the left side?
How does the distributive law work for simplifying $[(p\land q)\land r)]\lor [(p\land q)\land\lnot r)]$? Like this?
$(p\land q)$ multiplies $(p\land q)$ which gives the same result so this part is $(p\land q)$ with the $\land$ sign in the middle.
Then $r\lor\lnot r$ and final becomes $(p\land q) \land (r \lor \lnot r)$. This is correct but is this how it's supposed to be done?
- Last thing I'm confused on is the last part where you get $(p \lor \lnot q) \to s$ which is then finally simplified to $(q \to p) \to s$.
Is commutative law applied to $(p \lor \lnot q)$ then becomes $(\lnot q \lor p)$; then use conditional law?
Also, can I just stop at $(p \lor\lnot q) \to s$ OR do I have to simplify it one step more to $(q \to p) \to s$? IN OTHER WORDS, can I just ignore the implication/conditional law?
ALSO is writing the brackets required? for something like $(p \lor\lnot q) \to s$, can i write it without brackets like $p \lor \lnot q \to s$?
IMPORTANT: From each line of simplfication, where to write the law I used? Do I write it beside after using the law, or BEFORE using the law?
$\lnot\lnot p$ (use double negation)
$p$ ( double negation used)??
when to write the law used? Write it when literally using it, before using it, or after its used?