I wanted to find a truth function $f$ if it exists that make the formula below true:
$((p\to \lnot(q \oplus \lnot p)) \to (\lnot r \oplus (q \to p)))$
Where the $\oplus$ operator is defined as:
$(A \oplus B) \equiv ((A \land \lnot B) \lor (B \land \lnot A))$
My understanding was that to find the truth function that makes this formula true I needed to construct a truth table and the write down all rows where the LHS $\to $ RHS e.g $f(1,0,1) = 1$ However with the $\oplus$ operator definition the truth table would get quite length so I tried to simplify it with logical laws to make the formulas shorter for the LHS I simplified it down to $p \to \lnot q $ but for the RHS I had a bit more trouble with the logical laws and only managed to get it down to:
$((\lnot r \land (q \land \lnot p)) \lor (r \land (\lnot q \lor p))$
Am I on the right track with how I've attempted this question? Is it necessary to simplify the RHS further before putting it in a truth table?