[Math] Exclusive-NOR (XNOR) and his CNF sub-expression

logic

On Wikipedia, for example, we have the AND gate and its CNF sub-expression:

$(\neg A \lor \neg B \lor C) \wedge (A \lor \neg C) \wedge (B \lor \neg C)$

What is the CNF sub-expression for the Exclusive-NOR (XNOR) logic gate?

Best Answer

You can read a CNF off the truth table for XNOR: $C \leftrightarrow \overline{A\oplus B}$ exactly iff $$ A \land B \to C \\ A \land \bar B \to \bar C \\ \bar A \land B \to \bar C \\ \bar A \land \bar B \to C $$ which by unfolding the implications becomes $$ (\bar A \lor \bar B \lor C) \land \\ (\bar A \lor B \lor \bar C) \land \\ (A \lor \bar B \lor \bar C) \land \\ (A \lor B \lor C) $$

Related Question