Can i get the CNF of the following expression if i know the DNF?
I've the following expression:
$$\Bigl(\bigl(A\rightarrow (\overline A \land B) \bigr)\land \bigl((\overline A \land B)\rightarrow A\bigr)\Bigr)\rightarrow\bigl(B\land \overline B\bigr)$$
The DNF will be:
$A\rightarrow B \Rightarrow \overline A \lor B:$
$$\Rightarrow\Bigl(\bigl(\overline A \lor(\overline A \land B)\bigr)\land \bigl((A\lor\overline B)\lor A\bigr)\Bigr)\rightarrow \bigl(B\land \overline B\bigr)$$
$\overline A \lor(\overline A \land B) \Rightarrow \overline A$
$(A\lor\overline B)\lor A \Rightarrow A\lor\overline B$
$$\Rightarrow \bigl(\overline A\land (A\lor \overline B)\bigr)\rightarrow (B\land \overline B) $$
$A\rightarrow B \Rightarrow \overline A \lor B:$
$$\Rightarrow\overline{\bigl(\overline A \land(A\lor\overline B)\bigr)}$$
$$\Rightarrow \bigl( A\lor(\overline A\land B)\bigr)$$
$$\Rightarrow (A\lor B)$$
So, As we can see, The DNF of this expression is $A\lor B$, My question: Is it correct to say that the CNF of this expression will be $\overline A\land \overline B$?
I looked at Graham Kemp's Answer for this question, And didn't successfully understood how an expression that has only the operator $\land$ can be both conjunction of two disjunctions of one literal and disjunction of two conjunctions of one literal at the same time.
Because as i know (Please correct me if i wrong), A CNF has the form of $A\land B\land C\land D$ Where $A,B,C,D$ are expressions of the form $x\lor y\lor z$, And DNF has the form of $A\lor B\lor C\lor D$ Where $A,B,C,D$ are expressions of the form $x\land y\land z$.
Thanks for reading the question so far, please correct me if i made any mistake.
Thanks!!!