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

CHAP. 1] SET THEORY 5 Fig. 1-3 Recall that setsAandBare said to bedisjointornonintersectingif they have no elements in common or ...

6 SET THEORY [CHAP. 1 Fig. 1-4 Complements, Differences, Symmetric Differences Recall that all sets under consideration at a par ...

CHAP. 1] SET THEORY 7 We note that: (i) There arem= 2 nsuch fundamental products. (ii) Any two such fundamental products are dis ...

8 SET THEORY [CHAP. 1 Remark:Each law in Table 1-1 follows from an equivalent logical law. Consider, for example, the proof of D ...

CHAP. 1] SET THEORY 9 Proof.In counting the elements ofA∪B, first count those that are inA. There aren(A)of these. The only othe ...

10 SET THEORY [CHAP. 1 1.7Classes of Sets, Power Sets, Partitions Given a setS, we might wish to talk about some of its subsets. ...

CHAP. 1] SET THEORY 11 Fig. 1-6 EXAMPLE 1.11 Consider the following collections of subsets ofS={ 1 , 2 ,..., 8 , 9 }: (i) [{1, 3 ...

12 SET THEORY [CHAP. 1 1.8Mathematical Induction An essential property of the setN={1, 2, 3, ...} of positive integers follows: ...

CHAP. 1] SET THEORY 13 (c) C={x∈N| 4 +x= 3 } (a) Aconsists of the positive integers between 3 and 9; henceA={ 4 , 5 , 6 , 7 , 8 ...

14 SET THEORY [CHAP. 1 1.8 Prove Theorem 1.4. The following are equivalent:A⊆B,A∩B=A,A∪B=B. SupposeA⊆Band letx∈A. Thenx∈B, hence ...

CHAP. 1] SET THEORY 15 1.13 Determine the validity of the following argument: S 1 : All my friends are musicians. S 2 : John is ...

16 SET THEORY [CHAP. 1 (a) Find the number of people who read at least one of the three magazines. (b) Fill in the correct numbe ...

CHAP. 1] SET THEORY 17 CLASSES OF SETS 1.17 LetA=[{1, 2, 3}, {4, 5}, {6, 7, 8}]. (a) List the elements ofA;(b) Findn(A). (a) Aha ...

18 SET THEORY [CHAP. 1 1.23 Prove (De Morgan’s law): For any indexed class{Ai|i∈I}, we have (⋃ iAi )C = ⋂ iA C i. Using the defi ...

CHAP. 1] SET THEORY 19 SET OPERATIONS 1.29 Consider the universal setU={1, 2, 3, ..., 8, 9} and setsA={1, 2, 5, 6},B={2, 5, 7},C ...

20 SET THEORY [CHAP. 1 FINITE SETS AND THE COUNTING PRINCIPLE 1.39Determine which of the following sets are finite: (a) Lines pa ...

CHAP. 1] SET THEORY 21 INDUCTION 1.50Prove: 2+ 4 + 6 +···+ 2 n=n(n+ 1 ) 1.51Prove: 1+ 4 + 7 +···+ 3 n− 2 =n(^3 n 2 −^1 ) 1.52Pro ...

22 SET THEORY [CHAP. 1 Fig. 1-10 Fig. 1-11 1.41Use the data to fill in the Venn diagram in Fig. 1-11(b). Then: (a) 5; (b) 4; (c) ...

CHAPTER 2 Relations 2.1Introduction The reader is familiar with many relations such as “less than,” “is parallel to,” “is a subs ...

24 RELATIONS [CHAP. 2 Fig. 2-1 There are two things worth noting in the above examples. First of allA×B=B×A. The Cartesian prod ...