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

CHAP. 15] BOOLEAN ALGEBRA 407

Answers to Supplementary Problems

15.43. (a)a+a′b=a+b.

(b)a· 0 +a· 1 =a.

(c)ab+bc=(a+c)b.

15.45. (b)D 55 ; atoms 5 and 11. (d)D 130 ; atoms 2, 5 and 13.

15.46. (a) There are eight elements 1, 2, 5, 10, 11, 22, 55,

110. See Fig. 15-43(a).

(b) There are five subalgebras: { 1 , 110 },

{ 1 , 2 , 55 , 110 },{ 1 , 5 , 22 , 110 },{ 1 , 10 , 11 , 110 },

D 110.

(c) There are 15 sublattices which include the above

three subalgebras.

(d)A={ 2 , 5 , 11 }.

(e) See Fig. 15-43(b).

15.48. Maxterms: 30, 42, 70, 105

Fig. 15-43

15.49. (b)Hint: Use duality.

15.53. (a)xy′z; (b) 0; (c)xy′z′t; (d) 0.

15.54. (a)E=xy′+xy′z=xy′z′+xy′z.

(b)E=xy+xz′=xyz+xyz′+xy′z′.

(c)E=xy′+y′z=xy′z+xy′z′+x′y′z.

15.55. (a)E=xyz′+x′y′=xyz′+x′y′z+x′y′z′.

(b)E=x′y′=x′y′z+x′y′z′.

(c)E=x′yz′.

15.56. (a)Q=xzt. (b)Q=xyt′. (c) and (d) Does not

exist.

15.57. (a)EL= 11 ,ES=3; (b)EL= 11 ,ES=4.

15.58. (a)x′y, x′z′,y′z′.

(b)xy′,xzt′,y′zt′,x′z′t,y′z′t.

(c)xyzt, xz′t′,y′z′t′,x′y′z′,x′z′t.

15.59. (a)E=x′y+x′z′.

(b)E=xy′+xzt′+x′z′t+y′z′t.

(c)E=xyzt+xz′t′+x′y′z′+x′z′t.

15.60. (a)Y=A′BC+A′C′+BC′;

(b)ABC+A′BC+AB′C′.

15.61. (a)Y=(AB′)′+(A′+B+C)′+AC

(b)Y=(A′BC)′+A′BC′+(AB′C)′+AB′C′

15.62. See Fig. 15-44.

Fig. 15-44