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

388 BOOLEAN ALGEBRA [CHAP. 15

Abasic rectanglein a four-variable Karnaugh map is a square, two adjacent squares, four squares which form
a one-by-four or two-by-two rectangle, or eight squares which form a two-by-four rectangle. These rectangles
correspond to fundamental products with four, three, two, and one literal, respectively. Again, maximal basic
rectangles are the prime implicants. The minimization technique for a Boolean expressionE(x, y, z, t)is the
same as before.

EXAMPLE 15.18 Find the fundamental productPrepresented by the basic rectangle in the Karnaugh maps
shown in Fig. 15-23.
In each case, find the literals which appear in all the squares of the basic rectangle;Pis the product of such
literals.

(a) xy, andz′appear in both squares; henceP=xy′z′.

(b) Onlyyandzappear in all four squares; henceP=yz.

(c) Onlytappears in all eight squares; henceP=t.

Fig. 15-23

EXAMPLE 15.19 Use a Karnaugh map to find a minimal sum-of-products form for

E=xy′+xyz+x′y′z′+x′yzt′

Check all the squares representing each fundamental product. That is, check all four squares representing
xy′, the two squares representingxyz, the two squares representingx′y′z′, and the one square representing
x′yzt′, as in Fig. 15-24. A minimal cover of the map consists of the three designated maximal basic rectangles.

Fig. 15-24