Boolean algebra (logic)

Boolean algebra is the finitary algebra of two values. It resembles the algebra of real numbers as taught in high school, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the logical operations of conjunction xy, disjunction xy, and complement ¬x. The Boolean operations are these and all other operations obtainable from them by composition; equivalently, the finitary operations on the set {0,1}. The laws of Boolean algebra can be defined axiomatically as the equations derivable from a sufficient finite subset of those laws, such as the equations axiomatizing a complemented distributive lattice or a Boolean ring, or semantically as those equations identically true or valid over {0,1}. The axiomatic approach is sound and complete in the sense that it proves respectively neither more nor fewer laws than the validity-based semantic approach.