Boolean algebra

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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.

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Every law of Boolean algebra follows logically from these axioms. Furthermore, Boolean algebras can then be defined as the models of these axioms as treated in the section thereon. 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.

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This axiomatization woodes cc pentru opțiuni binare 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.

Or the intermediate notion of axiom can be sidestepped altogether by defining a Boolean law directly as any tautology, understood as an equation that holds for all values of its variables over 0 and 1.

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All these definitions of Boolean algebra can be shown to be equivalent. 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.

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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.

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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.

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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.

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The Duality Principle, also called De Morgan dualityasserts 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.

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We say that complement is a self-dual operation. The identity or do-nothing operation x copy the input to the output is also self-dual. There is no self-dual binary operation that depends on both its arguments.

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A composition of self-dual operations is a self-dual operation. The principle of woodes cc pentru opțiuni binare can be explained from a group theory perspective by the fact that there are exactly four functions that are one-to-one mappings automorphisms of the set of Boolean polynomials back to itself: the identity function, the complement function, the dual function and the contradual function complemented dual.

These four functions form a group under function compositionisomorphic to the Klein four-groupacting.

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