Reduce Logical Expression

Question

I have one week trying to solve this problem to no avail: $\lnot (p \lor q) \land r \lor \lnot (p \leftarrow \rightarrow (q \lor r))$

Wolfram says the answer is this: $(p \land \lnot q \land \lnot r) \lor ( \lnot p \land q) \lor ( \lnot p \land r)$, however I can only get to this:

$(\lnot q \land ((\lnot p \land r) \land (p \land \lnot r))) \lor ((q \lor r) \land \lnot p)$.

Any help will be appreciated.

Answer 1

Supposing that $a \land b \lor c \leftrightarrow (a \land b) \lor c$, then the DNF of your expression can be found this way.

$(\lnot (p \lor q) \land r) \lor \lnot (p \leftrightarrow (q \lor r) )$

$(\lnot p \land \lnot q \land r) \lor (p \oplus (q \lor r) )$

$(\lnot p \land \lnot q \land r) \lor (p \land \lnot (q \lor r) ) \lor (\lnot p \land (q \lor r) )$

$(\lnot p \land \lnot q \land r) \lor (p \land \lnot q \land \lnot r) \lor (\lnot p \land q) \lor (\lnot p \land r)$

And since $a \lor (a \land b) \leftrightarrow a$:

$(p \land \lnot q \land \lnot r) \lor (\lnot p \land q) \lor (\lnot p \land r)$