By Professor Dr. Peter L. Hammer, Professor Dr. Sergiu Rudeanu (auth.)

In classical research, there's a immense distinction among the category of difficulties which may be dealt with via the tools of calculus and the category of difficulties requiring combinatorial concepts. With the appearance of the electronic computing device, the excellence starts to blur, and with the expanding emphasis on difficulties regarding optimization over buildings, tIlE' contrast vanishes. what's useful for the analytic and computational therapy of vital questions coming up in smooth keep watch over thought, mathematical economics, scheduling thought, operations study, bioengineering, etc is a brand new and extra versatile mathematical thought which subsumes either the cla8sical non-stop and discrete t 19orithms. The paintings by means of HAMMER (IVANESCU) and RUDEANU on Boolean tools represents an incredible step during this dnectlOn, and it truly is hence an exceptional excitement to welcome it into print. it's going to definitely stimulate loads of extra study in either idea and alertness. RICHARD BELLMAN collage of Southern California FOf(,WOl'

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**Sample text**

Let (56) f(xI,""x n)= U c", ... OnXr' be the disjunctive canonical form of the function expression (51). Then (57) x,=~,f(pI, ... ,Pn)vfd(Pl, ... ,Pn) ... X~" f and let x, have the (i=I, ... ,n) by (55). Relations (51) and (57) maybe written in the concentrate form (58) x~,=c;~'f(pl""'PI/)vp~'r(pl" ,PIl) (i=I, ... ,n) 39 § 5. The Method of Bifurcations where IX, = (59) x~, ... x~n = ;~, .... ;~" f (PI ... , Pn) v p~, ... p~n (PI, ... , Pn)· 0 or 1. By multiplying the equalities (58), we obtain: 1 We deduce from (51), via (56) and (59), that U f(Xl, ...

N). The proof will use the disjunctive canonical form of the function and Lemma 5 below, which has an intrinsic interest. Lemma 5. The following identities hold: (53) }'n XiI . . x~n) V ( U by! Yn x~> ... na", ),, xi'· . x;,n) U a l' , ')'1- CU .. l', • . , ~ n ')'11' ,Yn Proof. Relation (53) is obvious. Identity (54) follows from the remark that y' =1= y" implies Xi,' x y" = XO Xl = X X = 0, while xi' xl' = xl' . Finally (55) follows by the DE MORGAN laws and the same remark. Proof of Theorem 11.

X~) of equation (15) can be written in the form (36), where (pT, ... ) if the vectors (pT, ... , p~) and (pT*, ... e. Xl* =1= X J** f or · (X t Ions, at least one j. We finish this section by giving an irredundant form of the general solution. Lemma 4. If relation a b = 0 holds, then an irredundant form of the general solution of equation (9) a X v bx= 0 is given by formula (43) x=bvp, together with the constraint (44) p~ali. Proof. Notice first that a b = 0 means b ~ a or b va = a. Now, relations (43) and (44) imply b::S;: b vp = X ~ b vali = b v a = a, hence they imply that X is a solution of equation (9), by Theorem 3.