Abstract

Algebraic reasoning in propositional logic and with sets is further abstracted in this optional chapter which studies the principles of Boolean algebras and their application in modelling computing systems through hardware design. A Boolean algebra is formally defined as a set with operations of join, meet and complementation as well as two distinguished elements 1 and 0 that satisfy a number of equational axioms. Three examples of Boolean algebra are presented to illustrate the axioms: the two-element Boolean algebra; the power set Boolean algebras; and the Boolean algebra of proposition. Next, fundamental laws of Boolean algebra, including absorption and De Morgan laws, are derived from the axioms by equational reasoning; and a duality principle is used to yield some of these theorems for free. Boolean algebra is then used for modelling basic logic gates (and gates, or gates and not gates), and it is shown how these elementary gates can be used to implement more complex circuitry, including half adders and full adders. It is also explored how Boolean algebra can play a role in implementing hardware circuits with a minimal numbers of basic gates.

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