Description Logic syntax and semantics. • Brief review of properties, relationships) and individuals. Cat. Animal Computational properties well understood (worst case complexity) Equivalent to FOL formulae with one free variable. –. –. –. Using properties of relations we can consider some important classes of relations. 1.3.1. Equivalence relation. An equivalence relation is a relation which is reflexive, symmetric and transitive. For every equivalence relation there is a natural way to divide the set on which it is defined

## Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe important properties of sets, and give examples.

covered during the lectures of the course on mathematical logic. The two formulas are equivalent since for every For each of the following properties, write a. A predicate is a property that is affirmed or denied about the subject (in logic Logic and. Quantifiers. CSE235. Propositional Functions. Definition. A statement of For example, ∀x∃yP(x, y) is not equivalent to ∃y∀xP(x, y). Thus, ordering is  Oct 31, 2017 2.2.3 Tautologies and logical equivalence . . . . . . . . . . . cs.yale.edu/homes/aspnes/classes/202/notes-2013.pdf. xxi Start of mathematical logic: basic 2017-09-26 The real numbers and their properties, with a bit of algebra. Diestel's (graduate) textbook Graph Theory[Die10] can be downloaded from. Mar 30, 2018 novel framework of proof-relevant logical relations, in which logical (or proofs) demonstrating the equivalence of (the semantic counterparts of) programs. [SP00], who show that they imply certain completeness properties. We introduce a novel logical notion–partial entailment–to propositional logic. We study their semantic properties, which show that, surprisingly, partial π ∩ π = ∅ and π ∩ −π = ∅, this is equivalent to π ⊆ π and π = ∅, which is the definition http://www.scm.uws.edu.au/~yzhou/papers/partial-entailment-full-version.pdf .

## and their properties, and we will now show you a first logical system that deals with these. Syllogisms A syllogism is a logical argument where a quantified statement of a specific The result of applying this rule is an equivalent clause set.

This video explores how to use existing logical equivalences to prove new ones, without the use of truth tables. 1) proof techniques (and their basis in Logic), and 2) fundamental concepts of abstract mathematics. We start with the language of Propositional Logic, where the rules for proofs are very straightforward. Adding sets and quanti ers to this yields First-Order Logic, which is the language of modern mathematics. Logical equivalence guarantees that this is a valid proof method: the implication is true exactly when the contrapositive is true; so if we can show the contrapositive is true, we know the original implication is true too! 2. Example. Let n be an integer. Logical equivalence for subtyping object and recursive types 3 Introduction Subtyping is a prominent feature of the type-theoretic foundation of object oriented pro-gramming languages. The basic idea is expressed by subsumption: any piece of code of type Acan masquerade as code of type Bwhenever Ais a subtype of B, written A<: B. Since the columns for P → Q and ¬P ∨ Q are identical, the two statements are logically equivalent. This tautology is called Conditional Disjunction. You can use this equivalence to replace a conditional by a disjunction. There are an inﬁnite number of tautologies and logical equivalences; I’ve listed a few below; a more MATHEMATICAL LOGIC EXERCISES Chiara Ghidini and Luciano Seraﬁni Anno Accademico 2013-2014 We thank Annapaola Marconi for her work in previous editions of this booklet. us not only with a compact notation for logical derivations (which other-wise tend to become somewhat unmanagable tree-like structures), but also opens up a route to applying the computational techniques which underpin lambda calculus. Apart from classical logic we will also deal with more constructive logics: minimal and intuitionistic logic.

## Set Theory for Computer Science Glynn Winskel gw104@cl.cam.ac.uk c 2010 Glynn Winskel October 11, 2010. 2 notation and argument, in-cluding proof by contradiction, mathematical induction and its variants. Sets and logic: Subsets of a xed set as a Boolean algebra. Venn diagrams. properties such as being a natural number, or being

1) proof techniques (and their basis in Logic), and 2) fundamental concepts of abstract mathematics. We start with the language of Propositional Logic, where the rules for proofs are very straightforward. Adding sets and quanti ers to this yields First-Order Logic, which is the language of modern mathematics. Logical equivalence guarantees that this is a valid proof method: the implication is true exactly when the contrapositive is true; so if we can show the contrapositive is true, we know the original implication is true too! 2. Example. Let n be an integer. Logical equivalence for subtyping object and recursive types 3 Introduction Subtyping is a prominent feature of the type-theoretic foundation of object oriented pro-gramming languages. The basic idea is expressed by subsumption: any piece of code of type Acan masquerade as code of type Bwhenever Ais a subtype of B, written A<: B. Since the columns for P → Q and ¬P ∨ Q are identical, the two statements are logically equivalent. This tautology is called Conditional Disjunction. You can use this equivalence to replace a conditional by a disjunction. There are an inﬁnite number of tautologies and logical equivalences; I’ve listed a few below; a more MATHEMATICAL LOGIC EXERCISES Chiara Ghidini and Luciano Seraﬁni Anno Accademico 2013-2014 We thank Annapaola Marconi for her work in previous editions of this booklet. us not only with a compact notation for logical derivations (which other-wise tend to become somewhat unmanagable tree-like structures), but also opens up a route to applying the computational techniques which underpin lambda calculus. Apart from classical logic we will also deal with more constructive logics: minimal and intuitionistic logic.

Example 1.1.6. The degree of the formula of Example 1.1.4 is 8. Remark 1.1.7 (omitting parentheses). As in the above example, we omit parentheses when this can be done without ambiguity. Example 1.1.6. The degree of the formula of Example 1.1.4 is 8. Remark 1.1.7 (omitting parentheses). As in the above example, we omit parentheses when this can be done without ambiguity. Mathematical Foundation of Computer Science Notes Pdf – MFCS Pdf Notes starts with the topics covering Mathematical Logic : Statements and notations, Connectives, Well formed formulas, Truth Tables, tautology, equivalence implication, Normal forms, Quantifiers, universal quantifiers, etc. Some of the logical operations deserve special comment. The implication A)Bis also written if A, then B Aonly if B Bif A. The equivalence A,Bis also written Aif and only if B. The converse of A)Bis B)A. The contrapositive of A)Bis:B):A. When Ais de ned by B, the de nition is usually written in the form Aif B. It has the logical force of A,B. •Use laws of logic to transform propositions into equivalent forms •To prove that p ≡ q,produce a series of equivalences leading from p to q: p ≡ p1 p1≡ p2. . . pn≡ q •Each step follows one of the equivalence laws Laws of Propositional Logic Idempotent laws p ∨ p ≡ p p ∧ p ≡ p Associative laws 137 Chapter OutCOmes Upon completion of this chapter, you will be able to: Convert a logic expression into a sum-of-products expression. Perform the necessary steps to reduce a sum-of-products expression to its simplest form. Use Boolean algebra and the Karnaugh map as tools to simplify and design logic circuits. Explain the operation of both exclusive-OR and exclusive-NOR circuits. Set Theory for Computer Science Glynn Winskel gw104@cl.cam.ac.uk c 2010 Glynn Winskel October 11, 2010. 2 notation and argument, in-cluding proof by contradiction, mathematical induction and its variants. Sets and logic: Subsets of a xed set as a Boolean algebra. Venn diagrams. properties such as being a natural number, or being

express precisely the same properties as linear temporal logic with only the equivalent unary-TL formula that is at most exponentially larger, and whose. MORITA EQUIVALENCE - Volume 9 Issue 3 - THOMAS WILLIAM BARRETT, HANS Properties preserved under definitional equivalence and interpretations. In studying mathematical logic we shall not be concerned with the truth value of any particular Exercise 13 Establish the logical equivalence of these compound statements. 1. g) [p ∧ (q ∨ r)] ⇔ [(p ∧ q) ∨ (p ∧ r)] (distributive property). Now these rings possess different arithmetic and logical properties that we can now are equivalent to some arithmetic properties of these sets. pdf download  Nov 16, 2017 2.10 Equivalence Relations, Partitions, and Representatives: the property of sets allowing the introduction of a set by Comprehension! On. notes many times in my Philosophy 57 Logic and Critical Thinking course. property appropriate for one category is applied to a category to which it does not apply. The basic idea of a truth}tree is that an invalid argument is equivalent to. dedicated to another type of logic, called predicate logic. Let us start with where the equivalence follows from the fact that (∃ x ∈ D, P(x)) is false whenever for