By Horand Störmer

ISBN-10: 3540528121

ISBN-13: 9783540528128

ISBN-10: 3642615198

ISBN-13: 9783642615191

In this ebook binary features and their illustration via implicants or implicates are defined. particularly minimum representations by means of leading implicants or top implicates are given. Such representations generalize the minimum representations of the standard Boolean capabilities. it truly is proven that implicants (implicates) of discrete services might be developed with the aid of implicants (implicates) of binary services. One titanic program is the outline of the reliability constitution of technical platforms, one other is using binary respectively discrete capabilities to categorise gadgets that are defined by way of the grades of yes attributes. ultimately a category of Boolean algebras of useful significance (set algebras, indicator algebras, algebras of sessions of propositions) are thought of. the weather of such algebras have representations that are strongly attached with the representations of binary functions.

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**Extra resources for Binary Functions and their Applications**

**Example text**

2 Theorem Let f be semi monotone and y E f(M). Let Gy • be the set of all minimal points and G; the set of all maximal points of {J = y}. 3) C,( {J = y}) = {l(z,z'l : (z, z') E Gy• x G;}. Thus a cube indicator is a prime implicant of l{f=y} if and only if the cube is an interval [z,z1 with a minimal point z of {f = y} and a maximal point z' of {J y}. = Proof a) Suppose C = 1(%,%'1 with (z, z') E G y• x G;. We show that C is a prime implicant of I{f=y}. The semi monotony of f implies that C is a prime implicant of l{f=y} with C < C'.

M;m·(r)), n·(r) = R(I*(f)), n;(r) = R(I;(f)) = n·(r) nI;(r). where S(Al, ... 14. Chapter 4 Reduction Methods In this chapter we state some rules which may be useful to determine the reduced set R(S(Al, ... ,Am» for any finite sets Al, ... 16). 14) with the help of this rules. 8 is suitable to determine the minimal sets (sets with minimal size) of R(S(Al, ... nts respectively (prime) implicates. We give the rules without proof. 1 Notation For each K with 0::/: K = {il, ... ) ~ N m , we define RK by RK := R(S(Aill ...

Maximal points of 1{/=2} are (2,4), (3, 2), (4,1). Prime implicants of 1{f=2} are 111 ,112' lIs, 114 ,116 ,118 with It = [1,2] x [3,4),12 = (2) x [2,4], 13 = [2,3] x [2] ,14 = [3] x [1,2], 55 Is = [3,4] x [1] ,16 = [4] x [0,1]. 4 we know that lU=2} has representations by prime implicates too. 3). We give such a circuit by the next figure. The proof is left to the reader. 20 suggest to introduce implicants and implicates of general discrete functions (see Davio et al. (1978)). Thus in this section we define implicants (prime implicants) of a discrete function.

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