Boolean Identities

The following identities directly follow the logical equivalences on slide 23 of Topic 3.

Name	Conjunction ($\boldsymbol{\wedge}$) version	Disjunction ($\boldsymbol{\vee}$) version
Identity Law	$x \wedge 1 = x$	$x \vee 0 = x$
Null Law	$x \wedge 0 = 0$	$x \vee 1 = 1$
Idempotent Law	$x \wedge x = x$	$x \vee x = x$
Complement Law	$x \wedge \neg x = 0$	$x \vee \neg x = 1$
Commutative Law	$x \wedge y = y \wedge x$	$x \vee y = y \vee x$
Associative Law	$(x \wedge y) \wedge z = x \wedge (y \wedge z)$	$(x \vee y) \vee z = x \vee (y \wedge z)$
Distributive Law	$x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$	$x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$
Absorption Law	$x \wedge (x \vee y ) = x$	$x \vee (x \wedge y) = x$
DeMorgan's Law	$\neg(x \wedge y) = \neg x \vee \neg y$	$\neg(x \vee y) = \neg x \wedge \neg y$
Double Negation Law	$\neg(\neg x) = \neg\neg x = x$