Lesson 3–5 Answers - Structured Independent Learning
... Lesson Questions Question 1 a) A relation is an association between two variables. Both graphs are relations. A function is a relation where for every x-value there is only one y-value. This is true for both graphs, therefore they are functions. b) Joe’s graph is discrete since there is no relation ...
... Lesson Questions Question 1 a) A relation is an association between two variables. Both graphs are relations. A function is a relation where for every x-value there is only one y-value. This is true for both graphs, therefore they are functions. b) Joe’s graph is discrete since there is no relation ...
Partiality and recursion in interactive theorem provers: An overview
... In the following, we introduce terminology and notational conventions that we use throughout this paper in the hope to ease its reading. 1.2.1. Terminology The term system refers to a proof assistant, including its logic and implementation. It defines the rules that specify which definitions and rea ...
... In the following, we introduce terminology and notational conventions that we use throughout this paper in the hope to ease its reading. 1.2.1. Terminology The term system refers to a proof assistant, including its logic and implementation. It defines the rules that specify which definitions and rea ...
The exponential function
... The most important and widely used exponential function has the particular base 2.7182818 . . . , a number always denoted by the single letter e: e = 2.7182818 . . . It will not be clear to the reader why this particular value is so important. However, its importance will become clear as your knowl ...
... The most important and widely used exponential function has the particular base 2.7182818 . . . , a number always denoted by the single letter e: e = 2.7182818 . . . It will not be clear to the reader why this particular value is so important. However, its importance will become clear as your knowl ...
Semantics of a Sequential Language for Exact Real
... on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination, as is usually done for discrete data types. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establi ...
... on the reals are infinite, one cannot decompose total correctness into the conjunction of partial correctness and termination, as is usually done for discrete data types. We instead introduce a suitable operational notion of strong convergence and show that total correctness can be proved by establi ...
Discrete Mathematics Study Center
... Since columns corresponding to p ∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r) match, the propositions are logically equivalent. This particular equivalence is known as the Distributive Law. The second method is to use a series of known logical equivalences to go from one propostion to the other Identity Law: p ∧ ...
... Since columns corresponding to p ∨ (q ∧ r) and (p ∨ q) ∧ (p ∨ r) match, the propositions are logically equivalent. This particular equivalence is known as the Distributive Law. The second method is to use a series of known logical equivalences to go from one propostion to the other Identity Law: p ∧ ...
Notes on Simply Typed Lambda Calculus
... Computational information in that they can be see as functional programming languages, or more realistically, a solid core on which to build a functional language. Logical information in two ways. First, typed λ-calculi can be used directly as logics—these are ‘intuitionistic type theories’ such as ...
... Computational information in that they can be see as functional programming languages, or more realistically, a solid core on which to build a functional language. Logical information in two ways. First, typed λ-calculi can be used directly as logics—these are ‘intuitionistic type theories’ such as ...
I. INTRODUCTION. ELEMENTS OF MATHEMATICAL LOGIC AND
... A formula which is true regardless of the truth values of its atomic subformulas is called tautology. First order logic (predicate calculus). Mathematical theories are expressed using first order logic. It differs from propositional logic by its use of quantified variables: (∀x ∈ A), (∀y ∈ R), (∀f1 ...
... A formula which is true regardless of the truth values of its atomic subformulas is called tautology. First order logic (predicate calculus). Mathematical theories are expressed using first order logic. It differs from propositional logic by its use of quantified variables: (∀x ∈ A), (∀y ∈ R), (∀f1 ...
Chapter 1: Sets, Functions and Enumerability
... c) Any subset S of P is enumerable. Use f(n) = n, n ∈ S; undefined otherwise. d) φ is enumerable. We can use the partial function e, whose domain is empty: undefined everywhere! Then the range of e is φ. e) The set of all people on earth is enumerable. Why? It’s finite. And any finite non-empty set, ...
... c) Any subset S of P is enumerable. Use f(n) = n, n ∈ S; undefined otherwise. d) φ is enumerable. We can use the partial function e, whose domain is empty: undefined everywhere! Then the range of e is φ. e) The set of all people on earth is enumerable. Why? It’s finite. And any finite non-empty set, ...
Chapter 13: Polynomials - Wayne State University
... We will not cover all there is to know about polynomials for the math competency exam. We will go over the addition, subtraction, and multiplication of polynomials. We will not go over factoring in this study guide. We will go over it in the study guide for PART B. So let’s begin with a few definiti ...
... We will not cover all there is to know about polynomials for the math competency exam. We will go over the addition, subtraction, and multiplication of polynomials. We will not go over factoring in this study guide. We will go over it in the study guide for PART B. So let’s begin with a few definiti ...
Q. 1 – Q. 5 carry one mark each.
... It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to completely pass a telegraph post. The length of the first train is 120 m and that of the second train is 150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________. (A ...
... It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to completely pass a telegraph post. The length of the first train is 120 m and that of the second train is 150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________. (A ...
Notes on First Order Logic
... If τ and τ 0 are terms, we write τ [x/τ 0 ] for the terms where x is replaced by τ 0 . We can formally define this operation by recursion: • x[x/τ 0 ] = τ 0 • y[x/τ 0 ] = y for x 6= y • c[x/τ 0 ] = c • F (τ1 , . . . , τn )[x/τ 0 ] = F (τ1 [x/τ 0 ], . . . , τn [x/τ 0 ]) The same notation can be used ...
... If τ and τ 0 are terms, we write τ [x/τ 0 ] for the terms where x is replaced by τ 0 . We can formally define this operation by recursion: • x[x/τ 0 ] = τ 0 • y[x/τ 0 ] = y for x 6= y • c[x/τ 0 ] = c • F (τ1 , . . . , τn )[x/τ 0 ] = F (τ1 [x/τ 0 ], . . . , τn [x/τ 0 ]) The same notation can be used ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.