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had been noticed treated further in axiomatizing boolean algebra 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 27 28 all these definitions of boolean algebra can be shown to be equivalent duality principle edit main article conjunction disjunction duality principle if x r is a partially ordered set then x r inverse is also a partially ordered set there is nothing special about the choice of symbols for the values of boolean algebra 0 and 1 could be renamed to α and β and as long as it was done consistently throughout it would still be boolean algebra albeit with some obvious cosmetic differences but suppose 0 and 1 were renamed 1 and 0 respectively then it would still be boolean algebra and moreover operating on the same values however it would not be identical to our original boolean algebra because now behaves the way used to do and vice versa so there are still some cosmetic differences to show that the notation has been changed despite the fact that 0s and 1s are still being used but if in addition to interchanging the names of the values the names of the two binary operations are also interchanged now there is no trace of what was done the end product is completely indistinguishable from what was started with the columns for x y and x y in the truth tables have changed places but that switch is immaterial when values and operations can be paired up in a way that leaves everything important unchanged when all pairs are switched simultaneously the members of each pair are called dual to each other thus 0 and 1 are dual and and are dual the duality principle also called de morgan duality asserts that boolean algebra is unchanged when all dual pairs are interchanged one change not needed to make as part of this interchange was to complement complement is a self dual operation the identity or do nothing operation x copy the input to the output is also self dual a more complicated example of a self dual operation is x y y z z x there is no self dual binary operation that depends on both its arguments a composition of self dual operations is a self dual operation for example if f x y z x y y z z x then f f x y z x t is a self dual operation of four arguments x y z t the principle of duality 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 composition isomorphic to the klein four group acting on the set of boolean polynomials walter gottschalk remarked that consequently a more appropriate name for the phenomenon would be the principle or square of quaternality 5 21 22 diagrammatic representations edit venn diagrams edit a venn diagram 29 can be used as a representation of a boolean operation using shaded overlapping regions there is one region for each variable all circular in the examples here the interior and exterior of region x corresponds respectively to the values 1 true and 0 false for variable x the shading indicates the value of the operation for each combination of regions with dark denoting 1 and light 0 some authors use the opposite convention the three venn diagrams in the figure below represent respectively conjunction x y disjunction x y and complement x figure 2 venn diagrams for conjunction disjunction and complement for conjunction the region inside both circles is shaded to indicate that x y is 1 when both variables are 1 the other regions are left unshaded to indicate that x y is 0 for the other three combinations the second diagram represents disjunction x y by shading those regions that lie inside either or both circles the third diagram represents complement x by shading the region not inside the circle 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 however we could put a circle for x in those boxes in which case each would denote a function of one argument x which returns the same value independently of x called a constant function as far as their outputs are concerned constants and constant functions are indistinguishable the difference is that a constant takes no arguments called a zeroary or nullary operation while a constant function takes one argument which it ignores and is a unary operation venn diagrams are helpful in visualizing laws the commutativity laws for and can be seen from the symmetry of the diagrams a binary operation that was not commutative would not have a symmetric diagram because interchanging x and y would have the effect of reflecting the diagram horizontally and any failure of commutativity would then appear as a failure of symmetry idempotence of and can be visualized by sliding the two circles together and noting that the shaded area then becomes the whole circle for both and to see the first absorption law x x y x start with the diagram in the middle for x y and note that the portion of the shaded area in common with the x circle is the whole of the x circle for the second absorption law x x y x start with the left diagram for x y and note that shading the whole of the x circle results in just the x circle being shaded since the previous shading was inside the x circle the double negation law can be seen by complementing the shading in the third diagram for x which shades the x circle to visualize the first de morgan s law x y x y start with the middle diagram for x y and complement its shading so that only the region outside both circles is shaded which is what the right hand side of the law describes the result is the same as if we shaded that region which is both outside the x circle and outside the y circle i e the conjunction of their exteriors which is what the left hand side of the law describes the second de morgan s law x y x y works the same way with the two diagrams interchanged the first complement law x x 0 says that the interior and exterior of the x circle have no overlap the second complement law x x 1 says that everything is either inside or outside the x circle digital logic gates edit digital logic is the application of the boolean algebra of 0 and 1 to electronic hardware consisting of logic gates connected to form a circuit diagram each gate implements a boolean operation and is depicted schematically by a shape indicating the operation the shapes associated with the gates for conjunction and gates disjunction or gates and complement inverters are as follows 30 from left to right and or and not gates the lines on the left of each gate represent input wires or ports the value of the input is represented by a voltage on the lead for so called active high logic 0 is represented by a voltage close to zero or ground while 1 is represented by a voltage close to the supply voltage active low reverses this the line on the right of each gate represents the output port which normally follows the same voltage conventions as the input ports complement is implemented with an inverter gate the triangle denotes the operation that simply copies the input to the output the small circle on the output denotes the actual inversion complementing the input the convention of putting such a circle on any port means that the signal passing through this port is complemented on the way through whether it is an input or output port the duality principle or de morgan s laws can be understood as asserting that complementing all three ports of an and gate converts it to an or gate and vice versa as shown in figure 4 below complementing both ports of an inverter however leaves the operation unchanged more generally one may complement any of the eight subsets of the three ports of either an and or or gate the resulting sixteen possibilities give rise to only eight boolean operations namely those with an odd number of 1s in their truth table there are eight such because the odd bit out can be either 0 or 1 and can go in any of four positions in the truth table there being sixteen binary boolean operations this must leave eight operations with an even number of 1s in their truth tables two of these are the constants 0 and 1 as binary operations that ignore both their inputs four are the operations that depend nontrivially on exactly one of their two inputs namely x y x and y and the remaining two are x y xor and its complement x y boolean algebras edit main article boolean algebra structure the term algebra denotes both a subject namely the subject of algebra and an object namely an algebraic structure whereas the foregoing has addressed the subject of boolean algebra this section deals with mathematical objects called boolean algebras defined in full generality as any model of the boolean laws we begin with a special case of the notion definable without reference to the laws namely concrete boolean algebras and then give the formal definition of the general notion concrete boolean algebras edit a concrete boolean algebra or field of sets is any nonempty set of subsets of a given set x closed under the set operations of union intersection and complement relative to x 5 historically x itself was required to be nonempty as well to exclude the degenerate or one element boolean algebra which is the one exception to the rule that all boolean algebras satisfy the same equations since the degenerate algebra satisfies every equation however this exclusion conflicts with the preferred purely equational definition of boolean algebra there being no way to rule out the one element algebra using only equations 0 1 does not count being a negated equation hence modern authors allow the degenerate boolean algebra and let x be empty example 1 the power set 2 x of x consisting of all subsets of x here x may be any set empty finite infinite or even uncountable example 2 the empty set and x this two element algebra shows that a concrete boolean algebra can be finite even when it consists of subsets of an infinite set it can be seen that every field of subsets of x must contain the empty set and x hence no smaller example is possible other than the degenerate algebra obtained by taking x to be empty so as to make the empty set and x coincide example 3 the set of finite and cofinite sets of integers where a cofinite set is one omitting only finitely many integers this is clearly closed under complement and is closed under union because the union of a cofinite set with any set is cofinite while the union of two finite sets is finite intersection behaves like union with finite and cofinite interchanged this example is countably infinite because there are only countably many finite sets of integers example 4 for a less trivial example of the point made by example 2 consider a venn diagram formed by n closed curves partitioning the diagram into 2 n regions and let x be the infinite set of all points in the plane not on any curve but somewhere within the diagram the interior of each region is thus an infinite subset of x and every point in x is in exactly one region then the set of all 2 2 n possible unions of regions including the empty set obtained as the union of the empty set of regions and x obtained as the union of all 2 n regions is closed under union intersection and complement relative to x and therefore forms a concrete boolean algebra again there are finitely many subsets of an infinite set forming a concrete boolean algebra with example 2 arising as the case n 0 of no curves subsets as bit vectors edit a subset y of x can be identified with an indexed family of bits with index set x with the bit indexed by x x being 1 or 0 according to whether or not x y this is the so called characteristic function notion of a subset for example a 32 bit computer word consists of 32 bits indexed by the set 0 1 2 31 with 0 and 31 indexing the low and high order bits respectively for a smaller example if x a b c displaystyle x a b c where a b c are viewed as bit positions in that order from left to right the eight subsets c b b c a a c a b and a b c of x can be identified with the respective bit vectors 000 001 010 011 100 101 110 and 111 bit vectors indexed by the set of natural numbers are infinite sequences of bits while those indexed by the reals in the unit interval 0 1 are packed too densely to be able to write conventionally but nonetheless form well defined indexed families imagine coloring every point of the interval 0 1 either black or white independently the black points then form an arbitrary subset of 0 1 from this bit vector viewpoint a concrete boolean algebra can be defined equivalently as a nonempty set of bit vectors all of the same length more generally indexed by the same set and closed under the bit vector operations of bitwise and as in 1010 0110 0010 1010 0110 1110 and 1010 0101 the bit vector realizations of intersection union and complement respectively prototypical boolean algebra edit main article two element boolean algebra the set 0 1 and its boolean operations as treated above can be understood as the special case of bit vectors of length one which by the identification of bit vectors with subsets can also be understood as the two subsets of a one element set this is called the prototypical boolean algebra justified by the following observation the laws satisfied by all nondegenerate concrete boolean algebras coincide with those satisfied by the prototypical boolean algebra this observation is proved as follows certainly any law satisfied by all concrete boolean algebras is satisfied by the prototypical one since it is concrete conversely any law that fails for some concrete boolean algebra must have failed at a particular bit position in which case that position by itself furnishes a one bit counterexample to that law nondegeneracy ensures the existence of at least one bit position because there is only one empty bit vector the final goal of the next section can be understood as eliminating concrete from the above observation that goal is reached via the stronger observation that up to isomorphism all boolean algebras are concrete boolean algebras the definition edit the boolean algebras so far have all been concrete consisting of bit vectors or equivalently of subsets of some set such a boolean algebra consists of a set and operations on that set which can be shown to satisfy the laws of boolean algebra instead of showing that the boolean laws are satisfied we can instead postulate a set x two binary operations on x and one unary operation and require that those operations satisfy the laws of boolean algebra the elements of x need not be bit vectors or subsets but can be anything at all this leads to the more gener...
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