I'm having trouble converting the below formula to disjunctive normal form using logical laws. I found the DNF using truth tables but I am having issues using just logical laws. Here is the formula:
$(A \to (A \land \lnot B)) \land (A \to (B \land \lnot A)))$
The DNF I found using truth tables:
$(\lnot A \land \lnot B) \lor (\lnot A \land B)$
Using the logical laws to get the DNF this was one of my attempts:
$(((\lnot A \lor ( A \land \lnot B)) \land (\lnot A \lor (B \land \lnot A)))$
few more steps..
arrived at : $((\lnot A \lor (A \land \lnot B))$ which is wrong
(1) $A \to (A \land \lnot B)=\lnot A \lor(A\land \lnot B)=\lnot A\lor \lnot B$
(2) $A \to (B \land \lnot A)=\lnot A \lor (B \land \lnot A)=\lnot A \lor B\land \lnot A=\lnot A \lor \lnot A \land B$
(3) $(\lnot A\lor \lnot B)\land (\lnot A \lor \lnot A \land B)=\lnot A \lor \lnot A\land B \lor \lnot B \land \lnot A=\lnot A$
The final result is $\lnot A$.