This article has multiple issues. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones. One may the algebra of random variables pdf this setup, allowing the algebra to be noncommutative.

This page was last edited on 5 September 2017, at 21:00. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations. In circuit engineering settings today, there is little need to consider other Boolean algebras, thus “switching algebra” and “Boolean algebra” are often used interchangeably. As with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.

4 possible combinations of inputs. 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. 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. 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.

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. 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. The end product is completely indistinguishable from what we started with.

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. There is no self-dual binary operation that depends on both its arguments. A composition of self-dual operations is a self-dual operation.

Boolean operation using shaded overlapping regions. There is one region for each variable, all circular in the examples here. 1 when both variables are 1. 0 for the other three combinations. While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a white box and a dark box, neither one containing a circle.