Analysis with an introduction to proof 5th by steven lay ch01

Chapter Logic and Proof Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 1.2, Slide 1-1 Section 1.2 Quantifiers Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 1.2, Slide 1-2 Is the sentence “x – 5x + = 0” a statement? No, as it stands it is not a statement because it is true for some values of x and false for others When a sentence involves a variable, we often refer to it using functional notation p(x): x – 5x + = For a specific value of x, p(x) becomes a statement that is true or false For example, p(2) is true and p(4) is false When a variable is used in an equation or an inequality, we assume that the general context for the variable is the set of real numbers unless told otherwise Within this context, we may remove the ambiguity of p(x) by using a quantifier The sentence “For every x, x – 5x + = 0” is a statement since it is false In symbols we write “∀ x, p(x).” The universal quantifier ∀ Copyright © 2013, 2005, 2001 Pearson Education, Inc is read “for every,” “for all,” “for each,” etc Section 1.2, Slide 1-3 The sentence “There exists an x such that x – 5x + = 0” is a statement and it is true In symbols we write “∃ x p(x).” The existential quantifier ∃ is read “there exists.” The symbol is an abbreviation for “such that.” Sometimes the quantifier is not explicitly written down, as in the statement “If x is greater than 1, then x is greater than 1.” The intended meaning is “∀ x, if x > 1, then x > 1.” In general, if a variable is used in the antecedent of an implication without being quantified, then the universal quantifier is assumed to apply Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 1.2, Slide 1-4 How we negate a quantified statement? Consider the statement “Everyone in the room is awake.” What condition must apply to the people in the room for this to be false? Must everyone be asleep? No, it is sufficient that at least one person be asleep On the other hand, in order for the statement “Someone in the room is asleep.” to be false, it must be the case that everyone is awake Symbolically, if p(x): x is awake then ~ [ ∀ x, p(x)] ⇔ [ ∃ x and ~ [∃ x Copyright © 2013, 2005, 2001 Pearson Education, Inc ~ p(x)] p(x)] ⇔ [ ∀ x, ~ p(x)] Section 1.2, Slide 1-5 Practice 1.2.3 Negate the following statements (b) “There exists a positive number y such that < g( y) ≤ 1.” Symbolically, we have “∃ y>0 < g( y) ≤ 1.” Note that < g( y) ≤ means < g( y) So the negation is and g( y) ≤ “ ∀ y > 0, g( y) ≤ or g( y) > 1.” In words we have “For every positive number y, g( y) ≤ or g( y) > 1.” (c) “For all x and y in A, there exists z in B such that x + y = z.” Note: This “and” is not used as a logical connective In symbols: “ ∀ x and y in A, ∃ z in B Copyright © 2013, 2005, 2001 Pearson Education, Inc x + y = z.” Section 1.2, Slide 1-6 Practice 1.2.3(d) Negate the following statement ∀ε >0∃N ∀ n, if n ≥ N, then ∀ x in S, | fn (x) – f (x)| < ε We work from left to right, negating each part as we go ∃ε >0 ~[∃N ∃ε >0 ∀ N ~ [ ∀ n, if n ≥ N, then ∀ x in S, | fn (x) – f (x)| < ε ] ∃ε >0 ∀N ∃n ∃ε >0 ∃ε >0 ∃ε >0 Copyright © 2013, 2005, 2001 Pearson Education, Inc ∀N ∃n ∀N ∃n ∀N ∃n ∀ n, if n ≥ N, then ∀ x in S, | fn (x) – f (x)| < ε ] ~ [ if n ≥ N, then ∀ x in S, | fn (x) – f (x)| < ε ] n ≥ N and ~ [ ∀ x in S, | fn (x) – f (x)| < ε ] n ≥ N and ∃ x in S n ≥ N and ∃ x in S ~ [ | fn (x) – f (x)| < ε ] | fn (x) – f (x)| ≥ ε ] Section 1.2, Slide 1-7 Caution: Take careful note of the order in which quantifiers are used Changing the order of two quantifiers can change the truth value For example, when talking about real numbers, the following statement is true ∀x∃y y > x Given any real number x, there is always a real number y that is greater than that x But the following statement is false ∃y ∀ x, y > x There is no fixed real number y that is greater than every real number Copyright © 2013, 2005, 2001 Pearson Education, Inc Section 1.2, Slide 1-8 ... > 1.” In general, if a variable is used in the antecedent of an implication without being quantified, then the universal quantifier is assumed to apply Copyright © 2013, 2005, 2001 Pearson Education,... which quantifiers are used Changing the order of two quantifiers can change the truth value For example, when talking about real numbers, the following statement is true ∀x∃y y > x Given any real... is true and p(4) is false When a variable is used in an equation or an inequality, we assume that the general context for the variable is the set of real numbers unless told otherwise Within this