Symbolic Logic I: The Propositional Calculus
... We also may apply the non-elementary logical operation ⇐⇒ to form F ⇐⇒ G, with the same understanding as above as to what this means. 5. Truth tables for general logical operations Since a general logical expression is built by successively applying elementary operations, its truth-value depends onl ...
... We also may apply the non-elementary logical operation ⇐⇒ to form F ⇐⇒ G, with the same understanding as above as to what this means. 5. Truth tables for general logical operations Since a general logical expression is built by successively applying elementary operations, its truth-value depends onl ...
Functions
... rule and a domain (the independent variable(s) x) for a relation or function, it is possible to determine all the ordered pairs that form the relation, which will determine the range (the dependent variable(s) y) of the relation. EXAMPLE: Find the range, R, of the relation defined by the rule 2x + 1 ...
... rule and a domain (the independent variable(s) x) for a relation or function, it is possible to determine all the ordered pairs that form the relation, which will determine the range (the dependent variable(s) y) of the relation. EXAMPLE: Find the range, R, of the relation defined by the rule 2x + 1 ...
Algebra IIa - Kalkaska Public Schools
... collected to establish that the Content and Skills have been learned? ...
... collected to establish that the Content and Skills have been learned? ...
Primitive Number Types
... Constants: final • A final variable is a constant • Once its value has been set, it cannot be changed • Named constants make programs easier to read and maintain • Convention: use all-uppercase names for constants final double QUARTER_VALUE = 0.25; final double DIME_VALUE = 0.1; final double NICKEL_ ...
... Constants: final • A final variable is a constant • Once its value has been set, it cannot be changed • Named constants make programs easier to read and maintain • Convention: use all-uppercase names for constants final double QUARTER_VALUE = 0.25; final double DIME_VALUE = 0.1; final double NICKEL_ ...
Document
... 1. All cats have tails 2. Tom is a cat 3. Tom has a tail Cannot conclude 3, given 1 and 2 using propositional logic ...
... 1. All cats have tails 2. Tom is a cat 3. Tom has a tail Cannot conclude 3, given 1 and 2 using propositional logic ...
Notes on Propositional and Predicate Logic
... • Simplify all subexpressions of the form (not (not p)) to p • Move all occurrences of or “inside” occurrences of and • Simplify all or- expressions for example by rewriting (or (or p q) r) as (or p q r) , and similarly for and Each premise is converted to conjunctive normal form in this way. Then t ...
... • Simplify all subexpressions of the form (not (not p)) to p • Move all occurrences of or “inside” occurrences of and • Simplify all or- expressions for example by rewriting (or (or p q) r) as (or p q r) , and similarly for and Each premise is converted to conjunctive normal form in this way. Then t ...
Set Concepts
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
Notes and exercises on First Order Logic
... • Some distinguished objects which will correspond to the constant symbols. • Some particular functions that correspond to the function symbols. • Some particular relations on the set of objects that correspond to the predicate symbols. The next definition makes all this precise: Definition 0.7 A st ...
... • Some distinguished objects which will correspond to the constant symbols. • Some particular functions that correspond to the function symbols. • Some particular relations on the set of objects that correspond to the predicate symbols. The next definition makes all this precise: Definition 0.7 A st ...
Year 9 Extended
... Linear inequalities, including their graphical representation. Number systems, rational and irrational numbers. Surds as one subset of irrational numbers. The definition of the absolute value of a number. Equations with absolute value. Absolute value inequalities. Quadratic equation. Quadratic formu ...
... Linear inequalities, including their graphical representation. Number systems, rational and irrational numbers. Surds as one subset of irrational numbers. The definition of the absolute value of a number. Equations with absolute value. Absolute value inequalities. Quadratic equation. Quadratic formu ...
Students extend their repertoire of functions to include polynomial
... ndards for matheematical practice describee practices that sstudents should develop p across grades in n their study of mathem matics. The Diagnostic Category Skkills List provvides description ns of skills that studentts can be expecteed to demonstrate within eeach Diagnostic C Category while ...
... ndards for matheematical practice describee practices that sstudents should develop p across grades in n their study of mathem matics. The Diagnostic Category Skkills List provvides description ns of skills that studentts can be expecteed to demonstrate within eeach Diagnostic C Category while ...
Section 1.5 Proofs in Predicate Logic
... Kronecker once said, “God made the integers, all else is the work of man.” In the late 1800s and early 1900s several intuitionists felt that the new theories which were sweeping mathematics at the time, like Cantor’s transfinite arithmetic, infinite sets, imaginary numbers, proof by contradictions, ...
... Kronecker once said, “God made the integers, all else is the work of man.” In the late 1800s and early 1900s several intuitionists felt that the new theories which were sweeping mathematics at the time, like Cantor’s transfinite arithmetic, infinite sets, imaginary numbers, proof by contradictions, ...
Truth, Conservativeness and Provability
... important requirement for deflationary truth theories. It is the deflationist’s intuition that truth is in some sense ‘innocent’ or ‘metaphysically thin’. 1 The truth predicate is just a ‘logical device’ permitting us to formulate useful generalizations (moreover, some of these generalizations will ...
... important requirement for deflationary truth theories. It is the deflationist’s intuition that truth is in some sense ‘innocent’ or ‘metaphysically thin’. 1 The truth predicate is just a ‘logical device’ permitting us to formulate useful generalizations (moreover, some of these generalizations will ...
The Closed World Assumption
... A theory T in predicate logic is a set of sentences. (We do not insist that T contain only definite clauses.) A theory T is said to be complete for ground atomic formulas if, for every ground atomic formula A, either T |= A or T |= ¬A. Let T be any theory (it does not have to be complete for ground ...
... A theory T in predicate logic is a set of sentences. (We do not insist that T contain only definite clauses.) A theory T is said to be complete for ground atomic formulas if, for every ground atomic formula A, either T |= A or T |= ¬A. Let T be any theory (it does not have to be complete for ground ...
Lecture 14 Notes
... calculate the value of a composed formula from values of the subformulas. In first-order logic, our starting point has to be atomic formulas instead of propositional variables and we have to explain how to calculate the value of quantified formulas that may have infinitely many subformulas. The stan ...
... calculate the value of a composed formula from values of the subformulas. In first-order logic, our starting point has to be atomic formulas instead of propositional variables and we have to explain how to calculate the value of quantified formulas that may have infinitely many subformulas. The stan ...
5.2 - Rational, Power, and Piecewise-Defined Functions
... When finding the domain of a piecewise-defined function, first check the domain intervals and see if any values of x do not have a rule identified. Then, check each function piece to make sure it is defined everywhere on that interval. Ex: Find the domain, using interval notation, for each of the fo ...
... When finding the domain of a piecewise-defined function, first check the domain intervals and see if any values of x do not have a rule identified. Then, check each function piece to make sure it is defined everywhere on that interval. Ex: Find the domain, using interval notation, for each of the fo ...
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.