This chapter includes contents: Uses a explicit linear ordering, insertions and removals are performed individually, allows insertions at both the front and back of the deque. This topic discusses the concept of a queue: Description of an Abstract Deque, applications, implementations, the STL and iterations. | (CSC 102) Lecture 8 Discrete Structures Previous Lectures Summary Predicates Set Notation Universal and Existential Statement Translating between formal and informal language Universal conditional Statements Equivalence Form Implicit Qualification Negations Predicates and Quantified statements II Today's Lecture Statements containing “∀ ” and “∃” Nested Quantifiers Relations Universal Instantiation statement Universal Modus Ponens Universal Modus Tollens Quantified form of Converse and Inverse error Multiple Quantified Statements Informally ∀ positive numbers x, ∃ a positive number y such that y < x ∃ a positive number x such that ∀ positive numbers y , y < x Sol: a. Given any positive number, there is another positive number that is smaller than the given number b. There is a positive number with the property that all positive numbers are smaller than this number. Every body loves some body Some body loves every body Any even integers equals twice some other integer There is a program that gives the correct answer to every question that is posed to it. Sol: a. ∀ people x, ∃ a person y such that x loves y. b. ∃ a person x such that ∀ people y , x loves y. c. ∀ even integers m, ∃ integers n, n = 2m. d. ∃ a program P such that ∀ questions it gives correct answer. Multiple Quantified Statements formally The negation of ∀ x, ∃ y such that P(x ,y) is logically equivalent to ∃ x such that ∀ y, ~P(x, y). A similar sequence of reasoning can be used to derive the following: The negation of ∃ x such that ∀ y, Q(x, y). is logically equivalent to ∀ x, ∃ y such that ~Q(x ,y). Negations of Multiple Statements Examples ∀ integers n, ∃ an integer k such that n = 2k. ∃ a person x such that ∀ people y, x loves y. Sol: a. ∃ an integer n such that ∀ integers k, Or we can say “ there is a some integer that is not even” b. ∀ people x, ∃ a person y such that x does not love y. Or we can say “ Nobody Loves everybody” Nested Quantifiers Two quantifiers are nested if one is within the . | (CSC 102) Lecture 8 Discrete Structures Previous Lectures Summary Predicates Set Notation Universal and Existential Statement Translating between formal and informal language Universal conditional Statements Equivalence Form Implicit Qualification Negations Predicates and Quantified statements II Today's Lecture Statements containing “∀ ” and “∃” Nested Quantifiers Relations Universal Instantiation statement Universal Modus Ponens Universal Modus Tollens Quantified form of Converse and Inverse error Multiple Quantified Statements Informally ∀ positive numbers x, ∃ a positive number y such that y < x ∃ a positive number x such that ∀ positive numbers y , y < x Sol: a. Given any positive number, there is another positive number that is smaller than the given number b. There is a positive number with the property that all positive numbers are smaller than this number. Every body loves some body Some body loves every body Any even integers equals twice some other integer There is a .
