By G.P. Gavrilov, A.A. Sapozhenko

Many years of useful event in educating discrete arithmetic shape the foundation of this article ebook. half I includes difficulties on such themes as Boolean algebra, *k*-valued logics, graphs and networks, parts of coding conception, automata concept, algorithms conception, combinatorics, Boolean minimization and logical layout. The workouts are preceded by means of plentiful theoretical heritage fabric. For additional examine the reader is talked about the large bibliography. half II follows an analogous constitution as half I, and offers useful tricks and recommendations. *Audience:*This ebook could be of serious worth to undergraduate scholars of discrete arithmetic, while the tougher workouts, which contain approximately one-third of the cloth, also will entice postgraduates and researchers.

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**Example text**

Let the function f( in) and g(fr) depend essentially on all their variables and let the variables Xl, ... , Xn, Yl, ... , Ym be pairwise different. Show that the function f(Xl, ... , Xn-l, g(Yl' ... , Ym)) depends essentially on all its variables. 12. Let pc(xn) be a set of all Boolean functions depending, and that too essentially, on the variables Xl, X2, ... , X", (1) Enumerate all functions in pC(X2). (2) Find the number IPC(X 3 )1. (3) Show that IPc(xn)1 == f=(_ll(~)22n-k. k=O (4) Show that lim 2- 2n IPC(X n)1 == 1.

We have l(x,x,y,O) = X V y. To obtain the required function, it remains to replace y by y. Thus, we have l(x, x, y, 0) = x V y. 1. Expanding the function 1 into the Zhegalkin polynomial, find out whether it is linear: (1) 1 = x --+ y; (2) 1 = X --+ Y EEl xy; (3) 1 = xy(x '" y); (4) 1 = xyV xyV z; (5) 1 = (xy V xy)z V z(xy V xy); (6) (7) 1 = ((x --+ y)(y --+ x)) '" z; 1 = xyz V xy; (8) 1 = xyz EEl xyz EEl xy; (9) 1 = m(x, y, z) EEl xyz EEl xyz; (10) 1 = (x V yz) EEl xyz; (11) 1 = (x V yz) EEl xyz; (12) 1 = (xyz V xyz) EEl x(y EEl z); (13) 1 = (xyz EEl x(yz) EEl x(y V z); (14) 1 = (xyz EEl zxy) V (xyz EEl xyz); (15) 1 = (xyz '" xyz) '" (xyz '" xyz).

1) A = {O, l,x}; (2) A = {x EB y,x rv y, I}; (3) A = {x, x ffi y, x ffi y ffi z}; (4) A = {xy,x Vy,xy V z}; (5)A={xVy,x-ty}; (6) A = {xy, xy }; (7) A = {x EB y EB z,xy V yz V zx,x}; (8) A = {l,x rv y,x EB y EB z EB I}; (9) A = {xy, xy V xz}; 40 CHAPTER 2. (10) A CLOSED CLASSES AND COMPLETENESS = {X,X Vy,x Vy V z,xy V z}. 5. Find out which of the below sets are closed classes: (1) the set of all functions of one variable; (2) the set of all functions of two variables; (3) the set of all functions f( Xl, X2, ...