study guide.
... • A predicate is like a propositional variable, but with free variables, and can be true or false depending on the value of these free variables. A domain of a predicate is a set from which the free variables can take their values (e.g., the domain of Even(n) can be integers). • Quantifiers For a pr ...
... • A predicate is like a propositional variable, but with free variables, and can be true or false depending on the value of these free variables. A domain of a predicate is a set from which the free variables can take their values (e.g., the domain of Even(n) can be integers). • Quantifiers For a pr ...
Book Question Set #1: Ertel, Chapter 2: Propositional Logic
... A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional formulas to be logically equivalent? If two propositional formulas are logically equivalent, they must evaluate to the same truth values for all interpretations. 7.) What does it mean for a logical for ...
... A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional formulas to be logically equivalent? If two propositional formulas are logically equivalent, they must evaluate to the same truth values for all interpretations. 7.) What does it mean for a logical for ...
Partial Correctness Specification
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
Document
... idea is that we can avoid commitment to R (the set of all sets that are not members of themselves) by arranging all sentences (or, equivalently, all propositional functions) into a hierarchy. The lowest level of this hierarchy will consist of sentences about individuals. The next lowest level will c ...
... idea is that we can avoid commitment to R (the set of all sets that are not members of themselves) by arranging all sentences (or, equivalently, all propositional functions) into a hierarchy. The lowest level of this hierarchy will consist of sentences about individuals. The next lowest level will c ...
CPS130, Lecture 1: Introduction to Algorithms
... p(1) is true and ((for all k1 since p(1) is
true ...
... p(1) is true and ((for all k
Midterm Topics for Midterm I
... Midterm I will be given on Wednesday, September 25, 2013, during the scheduled class time in the classroom, Science II, Room 207. The exam will be closed-note and closed-book, and no calculators will be permitted. Make sure to bring your student ID with you to the exam. Paper will be provided. ...
... Midterm I will be given on Wednesday, September 25, 2013, during the scheduled class time in the classroom, Science II, Room 207. The exam will be closed-note and closed-book, and no calculators will be permitted. Make sure to bring your student ID with you to the exam. Paper will be provided. ...
slides - National Taiwan University
... |= is about semantics, rather than syntax For Σ = ∅, we have ∅ |= τ , simply written |= τ . It says every truth assignment satisfies τ . In this case, τ is a tautology. ...
... |= is about semantics, rather than syntax For Σ = ∅, we have ∅ |= τ , simply written |= τ . It says every truth assignment satisfies τ . In this case, τ is a tautology. ...
p q
... Axioms (公設) An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms Given two distinct points, there is exactly one line that contains them. Given a line and a ...
... Axioms (公設) An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: Example: In Euclidean geometry the following are axioms Given two distinct points, there is exactly one line that contains them. Given a line and a ...
Exponentials and logarithms to the base e
... When we take derivatives and do integrals of exponentials and logarithms it is convenient if they are expressed in base e. This is similar to when we take derivatives and do integrals of trigonometric functions it is convenient if angles are expressed in radians. e is the number e = 2.718281828459… ...
... When we take derivatives and do integrals of exponentials and logarithms it is convenient if they are expressed in base e. This is similar to when we take derivatives and do integrals of trigonometric functions it is convenient if angles are expressed in radians. e is the number e = 2.718281828459… ...
Palo Alto 2016 - Stanford Introduction to Logic
... The above code checks to see if the first logical expression in the input string is satisfiable. If it is satisfiable, it adds that index (or the set of truth assignments that work) to an array for later use. The second for loop goes through the satisfied array, and checks if all those are satisfiab ...
... The above code checks to see if the first logical expression in the input string is satisfiable. If it is satisfiable, it adds that index (or the set of truth assignments that work) to an array for later use. The second for loop goes through the satisfied array, and checks if all those are satisfiab ...
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.