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

CHAP. 15] BOOLEAN ALGEBRA 399

(b) Herexy′z=00000100,xy=00000011, andz′=01010101. ThenE=xy′z+xy+z′=01010111. Thus

T( 00001111 , 00110011 , 01010101 )= 01010111

15.32. Find the truth tableT=T(E)for the Boolean expressionE=E(x, y, z)where:

(a) E=xyz′+x′yz; (b) E=xyz+xy′z+x′y′z.

HereEis a complete sum-of-products expression which is the sum of minterms. Example 15.13 gives the truth

tables for the minterms (using the special sequences). Each minterm contains a single 1 in its truth table; hence the

truth table ofEwill have 1’s in the same positions as the 1’s in the minterms inE. Thus:

(a) T (E)= 00001010 ; (b) T (E)= 01000101

15.33. Find the truth tableT=T(E)for the Boolean expression

E=E(x, y, z)=(x′y)′yz′+x′(yz+z′)

First expressEas a sum-of-products:

E=(x+y′)yz′+x′yz+x′z′=xyz′+y′yz′+x′yz+x′z′

=xyz′+x′yz+x′z′

Now expressEas a complete sum-of-products:

E=xyz′+x′yz+x′z′(y+y′)

=xyz′+x′yz+x′yz′+x′y′z′

As in Problem 15.32, use the truth tables for the minterms appearing in Example 15.13 to obtainT(E)=10101010.

15.34. Find the Boolean expressionE=E(x, y, z)corresponding to the truth table:

(a) T (E)= 01001001 ; (b) T (E)=00010001.

Each 1 inT(E)corresponds to the minterm with the 1 in the same position (using the truth tables for the minterms

appearing in Example 15.13). For example, the 1 in the second position corresponds tox′y′zwhose truth table has

a single 1 in the second position. ThenEis the sum of these minterms. Thus:

(a) E=x′y′z+x′yz+xyz′; (b) E=xy′z′+xyz

(Again we assume the input consists of the special sequences.)

KARNAUGH MAPS

15.35. Find the fundamental productPrepresented by each basic rectangle in the Karnaugh map in Fig. 15-31.

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

such literals.

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

(b) xandzappear in both squares; henceP=xz.

(c) Onlyzappears in all four squares; henceP=z.

Fig. 15-31