Free Boolean Algebra - A Simple Example

A Simple Example

The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John is tall" and "Mary is rich". These generate a Boolean algebra with four atoms, namely:

• John is tall, and Mary is rich;
• John is tall, and Mary is not rich;
• John is not tall, and Mary is rich;
• John is not tall, and Mary is not rich.

Other elements of the Boolean algebra are then logical disjunctions of the atoms, such as "John is tall and Mary is not rich, or John is not tall and Mary is rich". In addition there is one more element, FALSE, which can be thought of as the empty disjunction; that is, the disjunction of no atoms.

This example yields a Boolean algebra with 16 elements; in general, for finite n, the free Boolean algebra with n generators has 2n atoms, and therefore elements.

If there are infinitely many generators, a similar situation prevails except that now there are no atoms. Each element of the Boolean algebra is a combination of finitely many of the generating propositions, with two such elements deemed identical if they are logically equivalent.