Bridge to Abstract Mathematics: Mathematical Proof and Structures

3.1 BASIC CONCEPTS OF THE PREDICATE CALCULUS 83 a statement p(5): 5 > 4 which happens to be true. If 2 is substituted for x, ...

j 84 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 truth set of some propositional function, namely, the open sentence "x E P ...

3.1 BASIC CONCEPTS OF THE PREDICATE CALCULUS 85 The truth set of p(x) vq(x) equals (1, 3,4, 5,6, 79) = (1, 3,5,7,9) u {3,4, 5, ...

86 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 Solution We first calculate P and Q and express them in interval notation. T ...

3.1 BASIC CONCEPTS OF THE PREDICATE CALCULUS 87 theorems of set theory. You should understand, then, that our approach is to acc ...

88 LOGIC, PART 11: THE PREDICATE CALCULUS Chapter 3 Quantification It may have seemed surprising at the outset of Article 3.1 wh ...

3.2 QUANTIFICATION 89 Second, notice that the statement (Vx)(p(x)) is true precisely when the statement ha) is true for every po ...

I 90 LOGIC, PART It: THE PREDICATE CALCULUS Chapter 3 the word "some," and with sequences of words such as "there exists... such ...

3.2 QUANTIFICATION 91 need a substitution for n that makes r(n) true (n is a multiple of 4), where- as p(n) is false (n is not e ...

92 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 questions should be "yes." Before going on to the next article, try to decid ...

3.2 QUANTIFICATION 93 Translate into an English sentence each of the following statement forms: Let U be the set of all human b ...

94 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 Use your true-false answers from Exercise 3 to compare certain pairs of sta ...

3.3 THEOREMS ABOUT PREDICATES IN ONE VARIABLE 95 (c) r(x): x2 < 25 *(e) t(x): 3 < x < 5 (g) u(x): x2 I 1 (i) y(x):^0 &l ...

96 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 T H E 0 R E M 1 (Equivalences Involving Compound Predicates in One Variable) ...

3.3 THEOREMS ABOUT PREDICATES IN ONE VARIABLE 97 (or forces) that of the second. There are various other ways of expressing this ...

98 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 predicate -(p(x) + q(x)) is equivalent to (i.e., has the same truth set as) ...

3.3 THEOREMS ABOUT PREDICATES IN ONE VARIABLE 99 (c) The statement form (Vx)(p(x) v q(x)) from Theorem 2(a) asserts, when true, ...

100 LOGIC, PART II: THE PREDICATE CALCULUS Chapter 3 [Note: The concepts in (b) are from elementary linear algebra, (f) is from ...

3.4 QUANTIFICATION OF PROPOSITIONAL FUNCTIONS IN SEVERAL VARIABLES 161 [Hints: In (a) and (d), look at the logical negation of b ...

102 LOGIC, PART It: THE PREDICATE CALCULUS Chapter 3 Figure 3.1 Graphs pertaining to Example I. ...