Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

400 BOOLEAN ALGEBRA [CHAP. 15

15.36. LetRbe a basic rectangle in a Karnaugh map for four variablesx,y,z,t. State the number of literals in
the fundamental productPcorresponding toRin terms of the number of squares inR.
Pwill have one, two, three, or four literals according asRhas eight, four, two, or one squares.

15.37. Find the fundamental productPrepresented by each basic rectangleRin the Karnaugh map in Fig. 15-32.

In each case find those literals which appear in all the squares of the basic rectangle; thenPis the product of

such literals. (Problem 15.36 indicates the number of such literals inP.)

(a) There are two squares inR,soPhas three literals. Specifically,x′,y′,t′appear in both squares; henceP=x′y′t′.

(b) There are four squares inR,soPhas two literals. Specifically, onlyy′andtappear in all four squares; hence

P=y′t.

(c) There are eight squares inR,soPhas only one literal. Specifically, onlyyappears in all eight squares; hence

P=y.

Fig. 15-32

15.38. LetEbe the Boolean expression given in the Karnaugh map in Fig. 15-33.

(a) WriteEin its complete sum-of-products form. (b) Find a minimal form forE.

(a) List the seven fundamental products checked to obtain

E=xyz′t′+xyz′t+xy′zt+xy′zt′+x′y′zt+x′y′zt′+x′yz′t′

(b) The two-by-two maximal basic rectangle representsy′zsince onlyy′andzappear in all four squares. The

horizontal pair of adjacent squares representsxyz′, and the adjacent squares overlapping the top and bottom

edges representyz′t′. As all three rectangles are needed for a minimal cover,

E=y′z+xyz′+yz′t′

is the minimal sum forE.

15.39. Consider the Boolean expressionsE 1 andE 2 in variablesx,y,z,twhich are given by the Karnaugh maps
in Fig. 15-34. Find a minimal sum for (a)E 1 ;(b)E 2.

(a) Onlyy′appears in all eight squares of the two-by-four maximal basic rectangle, and the designated pair of

adjacent squares representsxzt′. As both rectangles are needed for a minimal cover,

E 1 =y′+xzt′

is the minimal sum forE 1.