6.2 Properties of operators

From the truth table definitions, you can conclude the following (by means of exhaustive truth table verification):

• Negation
  • double negation: $\neg(\neg(X)) = X$
• Conjunction
  • associative: $(X \wedge (Y \wedge Z)) = ((X \wedge Y) \wedge Z)$
  • commutative: $(X \wedge Y) = (Y \wedge X)$
• Disjunction
  • associative: $(X \vee (Y \vee Z)) = ((X \vee Y) \vee Z)$
  • commutative: $(X \vee Y) = (Y \vee X)$
• Combinations
  • distributive: $(X \wedge (Y \vee Z)) = (X \wedge Y) \vee (X \wedge Z)$
  • distributive: $(X \vee (Y \wedge Z)) = (X \vee Y) \wedge (X \vee Z)$
  • De Morgan's law: $\neg(X \wedge Y) = \neg(X) \vee \neg(Y)$
  • De Morgan's law: $\neg(X \vee Y) = \neg(X) \wedge \neg(Y)$
  • implication: $X \rightarrow Y = \neg X \vee Y$