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

CHAP. 14] ORDERED SETS AND LATTICES 365 14.33. See Fig. 14-21. Fig. 14-21 14.34. See Fig. 14-22. Fig. 14-22 14.35. See Fig. 14-2 ...

366 ORDERED SETS AND LATTICES [CHAP. 14 14.50. (a)e,f,g; (b) none; (c) sup(B)=e; (d) none. 14.51. (a) 1, 2, 3; (b) 8; (c) sup(A) ...

CHAP. 14] ORDERED SETS AND LATTICES 367 14.74. Geometrically, an elementa=Iis meet-irreducible if and only ifahas only one imme ...

CHAPTER 15 Boolean Algebra 15.1Introduction Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and ...

CHAP. 15] BOOLEAN ALGEBRA 369 Fig. 15-1 (b) LetBn=B×B×···×B(nfactors) where the operations of+,∗, and′are defined componentwise ...

370 BOOLEAN ALGEBRA [CHAP. 15 15.4Basic Theorems Using the axioms[B 1 ]through[B 4 ], we prove (Problem 15.5) the following theo ...

CHAP. 15] BOOLEAN ALGEBRA 371 (b) Consider the Boolean algebraD 70. Thenaprecedesbifadividesb. In such a case, lcm(a, b)=band gc ...

372 BOOLEAN ALGEBRA [CHAP. 15 Fig. 15-3 rectangle (universal set) into eight numbered sets which can be represented as follows: ...

CHAP. 15] BOOLEAN ALGEBRA 373 Definition 15.1:A Boolean expressionEis called asum-of-productsexpression ifEis a fundamental prod ...

374 BOOLEAN ALGEBRA [CHAP. 15 Step 4. The productxz′is contained inxyz′; hence, by the absorption law, xz′+(xz′y)=xz′ Thus we ma ...

CHAP. 15] BOOLEAN ALGEBRA 375 15.9Minimal Boolean Expressions, Prime Implicants There are many ways of representing the same Boo ...

376 BOOLEAN ALGEBRA [CHAP. 15 (b)P 1 =xy′andP 2 =y. Deletingyandy′yieldsQ=x. (c) P 1 =x′yzandP 2 =x′yt. No variable appears unco ...

CHAP. 15] BOOLEAN ALGEBRA 377 Fig. 15-7 EXAMPLE 15.9 We apply Algorithm 15.4 to the following expressionEwhich (by Example 15.8) ...

378 BOOLEAN ALGEBRA [CHAP. 15 Fig. 15-8 Suppose, for instance, the input data for the OR gate in Fig. 15-15(c)are the following ...

CHAP. 15] BOOLEAN ALGEBRA 379 Suppose, for instance, a NOT gate is asked to process the following three sequences: A 1 = 110001 ...

380 BOOLEAN ALGEBRA [CHAP. 15 EXAMPLE 15.10 Figure 15-12 is a typical AND-OR circuit with three inputs,A,B,Cand outputY. We can ...

CHAP. 15] BOOLEAN ALGEBRA 381 Fig. 15-14 15.11Truth Tables, Boolean Functions Consider a logic circuitLwithn=3 input devicesA,B, ...

382 BOOLEAN ALGEBRA [CHAP. 15 EXAMPLE 15.11 (a) Suppose a logic circuitLhasn=4 input devicesA,B,C,D. The 2n= 24 =16-bit special ...

CHAP. 15] BOOLEAN ALGEBRA 383 Boolean Functions LetEbe a Boolean expression withnvariablesx 1 ,x 2 ,...,xn. The entire discussio ...

384 BOOLEAN ALGEBRA [CHAP. 15 EXAMPLE 15.14 Find the sum of adjacent productsP 1 andP 2 where: (a) P=xyz′andP 2 =xy′z′. P 1 +P 2 ...