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

CHAP. 2] RELATIONS 25 EXAMPLE 2.3 (a) A=( 1 , 2 , 3 )andB={x, y, z}, and letR={( 1 , y), ( 1 , z), ( 3 ,y)}. ThenRis a relation ...

26 RELATIONS [CHAP. 2 Fig. 2-2 Directed Graphs of Relations on Sets There is an important way of picturing a relationRon a finit ...

CHAP. 2] RELATIONS 27 2.5Composition of Relations LetA,BandCbe sets, and letRbe a relation fromAtoBand letSbe a relation fromBto ...

28 RELATIONS [CHAP. 2 Composition of Relations and Matrices There is another way of findingR◦S. LetMRandMSdenote respectively th ...

CHAP. 2] RELATIONS 29 The relation (3) is not reflexive since no line is perpendicular to itself. Also (4) is not reflexive sinc ...

30 RELATIONS [CHAP. 2 EXAMPLE 2.9 (a) Determine which of the relations in Example 2.5 are transitive. The relationR 3 is not tra ...

CHAP. 2] RELATIONS 31 Transitive Closure LetRbe a relation on a setA. Recall thatR^2 =R◦RandRn=Rn−^1 ◦R. We define R∗= ⋃∞ i= 1 R ...

32 RELATIONS [CHAP. 2 (c) Letmbe a fixed positive integer. Two integersaandbare said to becongruent modulo m, written a≡b(modm) ...

CHAP. 2] RELATIONS 33 (b) LetR 5 be the relation of congruence modulo 5 on the setZof integers denoted by x≡y(mod 5) This means ...

34 RELATIONS [CHAP. 2 SolvedProblems PRODUCT SETS 2.1. Given:A={ 1 , 2 },B={x, y, z}, andC={ 3 , 4 }. Find:A×B×C. A×B×Cconsists ...

CHAP. 2] RELATIONS 35 Fig. 2-6 (c) Reverse the ordered pairs ofRto obtainR−^1 : R−^1 ={(y, 1 ), (z, 1 ), (y, 3 ), (x, 4 ), (z, 4 ...

36 RELATIONS [CHAP. 2 (b) The matrices ofMR,MS, andMR◦Sfollow: MR= 1 2 3 ⎡abc ⎣ 010 101 000 ⎤ ⎦ MS= a b c ⎡xyz ⎣ 010 100 011 ⎤ ⎦ ...

CHAP. 2] RELATIONS 37 TYPES OF RELATIONS AND CLOSURE PROPERTIES 2.9. Consider the following five relations on the setA={ 1 , 2 , ...

38 RELATIONS [CHAP. 2 (b) LetT=∩(S|Sis aP-relation andR⊆S). SincePisR-closable,Tis nonempty by (1) andTis aP-relation by (2). Si ...

CHAP. 2] RELATIONS 39 2.16. LetRbe the following equivalence relation on the setA={ 1 , 2 , 3 , 4 , 5 , 6 }: R={( 1 , 1 ), ( 1 , ...

40 RELATIONS [CHAP. 2 SupplementaryProblems RELATIONS 2.20. LetS={a, b, c},T={b, c, d}, andW={a, d}. FindS×T×W. 2.21. Findxandyw ...

CHAP. 2] RELATIONS 41 (d) IfRandSare reflexive thenR∪Sis reflexive. (e) IfRandSare transitive thenR∪Sis transitive. (f) IfRandSa ...

42 RELATIONS [CHAP. 2 2.26. (a) {( 9 , 1 ), ( 6 , 2 ), ( 3 , 3 )}; (b) (i) { 9 , 6 , 3 }, (ii){ 1 , 2 , 3 }, (iii){( 1 , 9 ), ( ...

CHAPTER 3 Functions and Algorithms 3.1Introduction One of the most important concepts in mathematics is that of a function. The ...

44 FUNCTIONS AND ALGORITHMS [CHAP. 3 Fig. 3-1 EXAMPLE 3.1 (a) Consider the functionf(x)=x^3 , i.e.,fassigns to each real number ...