Topic summary

Boolean algebras

Extracted from the Wikipedia article Boolean algebra (structure).

Examples

The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
- $(a \lor b) \land (\neg a \lor c) \land (b \lor c) \equiv (a \lor b) \land (\neg a \lor c)$
- $(a \land b) \lor (\neg a \land c) \lor (b \land c) \equiv (a \land b) \lor (\neg a \land c)$