rewriting logical statement form

[Hansel13]

I'm really stuck on this problem. I was able to apply these instructions to easier statements, but for some reason I can't even get started on this one. Any help?


Use the logical equivalences p ? q ? ~p V q and p ? q ? (~p V q) ^ (~q V p) to rewrite the given statement forms without using the symbol ? or ?.... AND use logical equivalence p V q = ~(~p ^ ~q) to rewrite each statement form using only ^ and ~.

Here is the statement: (p ? (q ? r)) ? ((p ^ q) ? r)

THANKS

 

[daon]

For the RHS of the iff:

\(\displaystyle (p \wedge q) \rightarrow r \equiv \neg (p \wedge q) \vee r \equiv \neg p \vee \neg q \vee r\)

LHS:

\(\displaystyle q \rightarrow r \equiv \neg q \vee r\)

\(\displaystyle p\rightarrow (q \rightarrow r) \equiv \neg p \vee \neg q \vee r\)

Now put them together and simplify if possible.

 

[Hansel13]

So we have:

(~p V ~q V r) <---> (~p V ~q V r)
= (~(~p V ~q V r) V (~p V ~q V r)) ^ (~(~p V ~q V r) V (~p V ~q V r))
= ((p ^ q ^ ~r) V (~p V ~q V r)) ^ ((p ^ q ^ ~r) V (~p V ~q V r))
= ((p ^ q ^ ~r) V (~(p ^ q) V r)) ^ ((p ^ q ^ ~r) V (~(p ^ q) V r))
= ((p ^ q ^ ~r) V ~((p ^ q) ^ ~r)) ^ ((p ^ q ^ ~r) V ~((p ^ q) ^ ~r))
= ~(~(p ^ q ^ ~r) ^ ((p ^ q) ^ ~r)) ^ ~(~(p ^ q ^ ~r) ^ ((p ^ q) ^ ~r))

Does this look correct. And can I simplify the end anymore?

thanks