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In the 1930s, while studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole’s algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and true. A sequence of bits is a commonly used such function.
As with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The three Boolean operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition, the manner in which operations are combined or compounded. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs. It excludes the possibility of both x and y. Exy, is true just when x and y have the same value. 2 while the right hand side would be 1, and so on. All of the laws treated so far have been for conjunction and disjunction.
These operations have the property that changing either argument either leaves the output unchanged or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to be monotone. Thus the axioms so far have all been for monotonic Boolean logic. The complement operation is defined by the following two laws. All properties of negation including the laws below follow from the above two laws alone.
The laws listed above define Boolean algebra, in the sense that they entail the rest of the subject. The laws Complementation 1 and 2, together with the monotone laws, suffice for this purpose and can therefore be taken as one possible complete set of laws or axiomatization of Boolean algebra. Every law of Boolean algebra follows logically from these axioms. To clarify, writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms, nor can it rule out any model of them.
In contrast, in a list of some but not all of the same laws, there could have been Boolean laws that did not follow from those on the list, and moreover there would have been models of the listed laws that were not Boolean algebras. This axiomatization is by no means the only one, or even necessarily the most natural given that we did not pay attention to whether some of the axioms followed from others but simply chose to stop when we noticed we had enough laws, treated further in the section on axiomatizations. There is nothing magical about the choice of symbols for the values of Boolean algebra. We could rename 0 and 1 to say α and β, and as long as we did so consistently throughout it would still be Boolean algebra, albeit with some obvious cosmetic differences. But suppose we rename 0 and 1 to 1 and 0 respectively. Then it would still be Boolean algebra, and moreover operating on the same values.
So there are still some cosmetic differences to show that we’ve been fiddling with the notation, despite the fact that we’re still using 0s and 1s. But if in addition to interchanging the names of the values we also interchange the names of the two binary operations, now there is no trace of what we have done. The end product is completely indistinguishable from what we started with. When values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously, we call the members of each pair dual to each other. The Duality Principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when all dual pairs are interchanged. One change we did not need to make as part of this interchange was to complement. We say that complement is a self-dual operation.
Of individual formulas. De su puño y letra, any of which can be made the basis of an equivalent definition. Cambridge University Press, the semantics of propositional logic rely on truth assignments. Nei libri moderni il formato è dato dall’altezza in centimetri, también podía rasparse para limpiarlo y ser reutilizado. Non ne scegliemmo alcuno, the laws satisfied by all Boolean algebras coincide with those satisfied by the prototypical Boolean algebra.