Rules of Boolean Algebra

Boolean Algebra satisfies several important algebraic properties that govern its operations. Let $x$, $y$, and $z$ be two Boolean variables. We see that:

• Commutativity holds, so \( x \land y = y \land x \), and \( x \lor y = y \lor x \).
• Associativity holds, so \( (x \land y) \land z = x \land (y \land z) \).
• Distributivity also holds, so $x \land (y \lor z) =$$\; (x \land y) \lor (x \land z)$.
• There are identity elements, where \( x \land 1 = x \), and \( x \lor 0 = x \).
• Boolean Algebra also includes complement laws that relate elements with their negations: for any element \( x \), we have \( x \lor \neg x = 1 \).
• Similarly, \( x \land \neg x = 0 \), and this expresses the law of contradiction.
• Idempotent laws hold, so \( x \lor x = x \), and \( x \land x = x \).
• Absorption laws also hold, such as \( x \lor (x \land y) = x \).