CHAP. 15] BOOLEAN ALGEBRA 385
This can be solved by using Karnaugh maps as follows:(a) Check the squares corresponding toxyandxy′as in Fig. 15-17(a). Note thatE 1 consists of one prime
implicant, the two adjacent squares designated by the loop in Fig. 15-17(a). This pair of adjacent squares
represents the variablex,soxis a (the only) prime implicant ofE 1. Consequently,E 1 =xis its minimal
sum.
Fig. 15-17(b) Check the squares corresponding toxy,x′y, andx′y′as in Fig. 15-17(b). Note thatE 2 contains two pairs of
adjacent squares (designated by the two loops) which include all the squares ofE 2 .The vertical pair represents
yand the horizontal pair representsx′; henceyandx′are the prime implicants ofE 2. ThusE 2 =x′+yis
its minimal sum.
(c) Check the squares corresponding toxyandx′y′as in Fig. 15-17(c). Note thatE 3 consists of two isolated
squares which representxyandx′y′; hencexyandx′y′are the prime implicants ofE 3 andE 3 =xy+x′y′
is its minimal sum.
Case of Three Variables
The Karnaugh map corresponding to Boolean expressionsE=E(x, y, z)with three variablesx,y,zis
shown in Fig. 15-18(a). Recall that there are exactly eight minterms with three variables:
xyz, xyz′,xy′z′,xy′z, x′yz, x′yz′,x′y′z′,x′y′zThese minterms are listed so that they correspond to the eight squares in the Karnaugh map in the obvious way.
Furthermore, in order that every pair of adjacent products in Fig. 15-18(a)are geometrically adjacent, the
right and left edges of the map must be identified. This is equivalent to cutting out, bending, and gluing the
map along the identified edges to obtain the cylinder pictured in Fig. 15-18(b)where adjacent products are now
represented by squares with one edge in common.
Fig. 15-18Viewing the Karnaugh map in Fig. 15-18(a)as a Venn diagram, the areas represented by the variablesx,y,
andzare shown in Fig. 15-19. Specifically, the variablexis still represented by the points in the upper half of
the map, as shaded in Fig. 15-19(a), and the variableyis still represented by the points in the left half of the
map, as shaded in Fig. 15-19(b). The new variablezis represented by the points in the left and right quarters of