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

CHAP. 15] BOOLEAN ALGEBRA 385 This can be solved by using Karnaugh maps as follows: (a) Check the squares corresponding toxyandx ...

386 BOOLEAN ALGEBRA [CHAP. 15 Fig. 15-19 the map, as shaded in Fig. 15-19(c). Thusx′,y′, andz′are represented, respectively, by ...

CHAP. 15] BOOLEAN ALGEBRA 387 (c) Check the squares corresponding to the five summands as in Fig. 15-20(c). As indicated by the ...

388 BOOLEAN ALGEBRA [CHAP. 15 Abasic rectanglein a four-variable Karnaugh map is a square, two adjacent squares, four squares wh ...

CHAP. 15] BOOLEAN ALGEBRA 389 The two-by-two squares represent the fundamental productsxzandy′z′, and the two adjacent squares ( ...

390 BOOLEAN ALGEBRA [CHAP. 15 Fig. 15-25 15.4. Find the number of subalgebras ofD 210. A subalgebra ofD 210 must contain two, fo ...

CHAP. 15] BOOLEAN ALGEBRA 391 Also, using the absorption law in the last step, a+R=a+(a∗(b∗c))=(a+a)∗(a+(b∗c))=a∗(a+(b∗c))=a Thu ...

392 BOOLEAN ALGEBRA [CHAP. 15 15.8. Prove Theorem 15.5: The following are equivalent in a Boolean algebra: ( 1 )a+b=b; ( 2 )a∗b= ...

CHAP. 15] BOOLEAN ALGEBRA 393 BOOLEAN EXPRESSIONS 15.10. Reduce the following Boolean products to either 0 or a fundamental prod ...

394 BOOLEAN ALGEBRA [CHAP. 15 15.15. LetE=xy′+xyz′+x′yz′. Prove that(a) xz′+E=E;(b) x+E=E;(c)z′+E=E. Since the complete sum-of ...

CHAP. 15] BOOLEAN ALGEBRA 395 15.19. Prove Lemma 15.10: SupposeQis the consensus ofP 1 andP 2. ThenP 1 +P 2 +Q=P 1 +P 2. Since t ...

396 BOOLEAN ALGEBRA [CHAP. 15 Fig. 15-26 Fig. 15-27 15.23. Express the outputYas a Boolean expression in the inputsA,B,Cfor the ...

CHAP. 15] BOOLEAN ALGEBRA 397 15.25. Express the outputYas a Boolean expression in the inputsAandBfor the logic circuit in Fig. ...

398 BOOLEAN ALGEBRA [CHAP. 15 The outputY=1 for an AND gate if and only if there are 1’s in all the positions of the input seque ...

CHAP. 15] BOOLEAN ALGEBRA 399 (b) Herexy′z=00000100,xy=00000011, andz′=01010101. ThenE=xy′z+xy+z′=01010111. Thus T( 00001111 , 0 ...

400 BOOLEAN ALGEBRA [CHAP. 15 15.36. LetRbe a basic rectangle in a Karnaugh map for four variablesx,y,z,t. State the number of l ...

CHAP. 15] BOOLEAN ALGEBRA 401 Fig. 15-33 Fig. 15-34 (b) The four corner squares form a two-by-two maximal basic rectangle which ...

402 BOOLEAN ALGEBRA [CHAP. 15 (a) Check the two squares corresponding to each ofx′yzandy′zt′, and check the square corresponding ...

CHAP. 15] BOOLEAN ALGEBRA 403 SupplementaryProblems BOOLEAN ALGEBRAS 15.43. Write the dual of each Boolean expression: (a) a(a′+ ...

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) ...