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

404 BOOLEAN ALGEBRA [CHAP. 15

15.56. Find the consensusQof the fundamental productsP 1 andP 2 where:

(a)P 1 =xy′z, P 2 =xyt; (c)P 1 =xy′zt, P 2 =xyz′;

(b)P 1 =xyz′t′,P 2 =xzt′; (d)P 1 =xy′t,P 2 =xzt.

15.57. For any Boolean sum-of-products expressionE,weletELdenote the number of literals inE(counting multiplicity)

andESdenote the number of summands inE. FindELandESfor each of the following:

(a)E=xyz′t+x′yt+xy′zt; (b)E=xyzt+xt′+x′y′t+yt.

15.58. Apply the consensus method (Algorithm 15.3) to find the prime implicants of each Boolean expression:

(a) E 1 =xy′z′+x′y+x′y′z′+x′yz;

(b) E 2 =xy′+x′z′t+xyzt′+x′y′zt′;

(c) E 3 =xyzt+xyz′t′+xz′t′+x′y′z′+x′yz′t.

15.59. Find a minimal sum-of-products form for each of the Boolean expressions in Problem 15.58.

LOGIC GATES,TRUTH TABLES

15.60. Express the outputYas a Boolean expression in the inputsA,B,Cfor the logic circuit in:

(a) Fig. 15-38(a); (b) Fig. 15-38(b).

Fig. 15-38

15.61. Express the outputYas a Boolean expression in the inputsA,B,Cfor the logic circuit in:

(a) Fig. 15-39(a); (b) Fig. 15-39(b).

15.62. Draw the logic circuitLwith inputsA,B,Cand outputYwhich corresponds to each Boolean expression:

(a)Y=AB′C+AC′+A′C; (b)Y=A′BC+A′BC′+ABC′.

15.63. Find the output sequenceYfor an AND gate with inputsA,B,C(or equivalently forY=ABC) where:

(a) A=110001;B=101101;C=110011.

(b) A=01111100;B=10111010;C=00111100.

(c) A=00111110;B=01111100;C=11110011.