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

CHAP. 3] FUNCTIONS AND ALGORITHMS 65 Fig. 3-10 MISCELLANEOUS PROBLEMS 3.22. Find the domainDof each of the following real-valued ...

66 FUNCTIONS AND ALGORITHMS [CHAP. 3 3.26. Prove the following equivalent formulation of the Schroeder–Bernstein Theorem 3.5: Su ...

CHAP. 3] FUNCTIONS AND ALGORITHMS 67 ONE-TO-ONE, ONTO, AND INVERTIBLE FUNCTIONS 3.30. Determine if each function is one-to-one. ...

68 FUNCTIONS AND ALGORITHMS [CHAP. 3 CARDINAL NUMBERS 3.43. Find the cardinal number of each set: (a){x|xis a letter in “BASEBAL ...

CHAP. 3] FUNCTIONS AND ALGORITHMS 69 3.34. f−^1 (x)=(x+ 7 )/ 3 3.35. (a) 425631; (b) 416253; (c) 534261; (d) 415623; (e) 453261. ...

CHAPTER 4 Logic and Propositional Calculus 4.1Introduction Many algorithms and proofs use logical expressions such as: “IFpTHENq ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 71 The fundamental property of a compound proposition is that its truth value is compl ...

72 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 Definition 4.2: Ifpandqare false, thenp∨qis false; otherwisep∨qis true. The truth v ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 73 The main property of a propositionP(p,q,...)is that its truth value depends exclusi ...

74 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 Fig. 4-4 4.5Tautologies and Contradictions Some propositionsP(p,q,...)contain onlyT ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 75 Fig. 4-6 4.7Algebra of Propositions Propositions satisfy various laws which are lis ...

76 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 In other words, the conditional statement “Ifpthenq” is logically equivalent to the ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 77 Fig. 4-8 That is, the following argument is valid: p→q, q→r$p→r(Law of Syllogism) T ...

78 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 Frequently, whenAis some set of numbers, the conditionp(x)has the form of an equati ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 79 which reads “There exists anxinAsuch thatp(x)is a true statement” or, simply, “For ...

80 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 EXAMPLE 4.10 (a) The following statements are negatives of each other: “For all pos ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 81 Propositional Functions with more than One Variable A propositional function (ofnva ...

82 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 (b) The formal definition thatLis the limit of a sequencea 1 ,a 2 ,...follows: ∀∈&g ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 83 4.5. Use the laws in Table 4-1 to show that¬(p∧q)∨(¬p∧q)≡¬p. Statement Reason ( 1 ) ...

84 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 (b) Note that¬(p↔q)≡p↔¬q≡¬p↔q; hence the negation of the statement is either of the ...