Simplicify $((A ⇒ B) ⇒ (B ⇒ A)) ⇒( ¬(A∧B) ⇔ ¬(B∨A))$

Question

I am trying to simplicify the expression:

$((A ⇒ B) ⇒ (B ⇒ A)) ⇒( ¬(A∧B) ⇔ ¬(B∨A))$

and stuck here. The definition of $\land$ is give n by $A\land B$ is $\lnot (A\implies \lnot B)$.

I tried to remove all the $\land$ and $\lor$, but it seems not work well and become messy. May I please what is the easiest way to go about something like this? Thanks in advance.

Answer 1

$A \implies B$ is logically equivalent to $\lnot A \lor B$. Therefore: $\lnot(\lnot(\lnot A\lor B)\lor(\lnot B\lor A))\lor(\lnot\lnot(A\land B)\lor\lnot(B\lor A))\land(\lnot\lnot(B\lor A)\lor\lnot(A\land B)).$

simplifying and drawing up a truth table the expression is reduced to $B\lor\lnot A$.

Answer 2

$$((A \Rightarrow B) \Rightarrow (B \Rightarrow A)) \Rightarrow( \neg (A\land B) \Leftrightarrow \neg (B\lor A)) = \text{ (Equivalence)}$$

$$((A \Rightarrow B) \Rightarrow (B \Rightarrow A)) \Rightarrow (( \neg (A\land B) \Rightarrow \neg (B\lor A)) \land ( \neg (B\lor A) \Rightarrow \neg (A\land B))) = \text{ (Implication)}$$

$$\neg (\neg (\neg A \lor B) \lor (\neg B \lor A)) \lor ( ( \neg \neg (A\land B) \lor \neg (B\lor A)) \land ( \neg \neg (B\lor A) \lor \neg (A\land B))) = \text{ (DeMorgan)}$$

$$(\neg \neg (\neg A \lor B) \land \neg (\neg B \lor A)) \lor ( ( \neg \neg (A\land B) \lor (\neg B\land \neg A)) \land ( \neg \neg (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (DeMorgan)}$$

$$(\neg \neg (\neg A \lor B) \land (\neg \neg B \lor \neg A)) \lor ( ( \neg \neg (A\land B) \lor (\neg B\land \neg A)) \land ( \neg \neg (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (Double Negation)}$$

$$((\neg A \lor B) \land (B \lor \neg A)) \lor ( ( (A\land B) \lor (\neg B\land \neg A)) \land ( (B\lor A) \lor (\neg A \lor \neg B))) = \text{ (Commutation, Association)}$$

$$((\neg A \lor B) \land (\neg A \lor B)) \lor ( ( (A\land B) \lor (\neg A\land \neg B)) \land ( A \lor \neg A \lor B \lor \neg B)) = \text{ (Complement)}$$

$$((\neg A \lor B) \land (\neg A \lor B)) \lor ( ( (A\land B) \lor (\neg A\land \neg B)) \land ( \top \lor \top)) = \text{ (Idempotence)}$$

$$(\neg A \lor B) \lor ( ( (A\land B) \lor (\neg A\land \neg B)) \land \top) = \text{ (Identity)}$$

$$(\neg A \lor B) \lor (A\land B) \lor (\neg A\land \neg B) = \text{ (Adjacency)}$$

$$B \lor (\neg A\land \neg B) = \text{ (Reduction)}$$

$$B \lor \neg A$$