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LogicPredicatesInduction

Combinatorics

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LogicMathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Topically, mathematical logic bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

Set theorySet theory is the branch of mathematical logic that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

PropositionThe term proposition has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false.

ContrapositionContraposition is a law that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped

ConverseThe converse of a categorical or implicational statement is the result of reversing its two parts.

InverseAn inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. Any conditional sentence has an inverse: the contrapositive of the converse.

SyllogismA syllogism (Greek: συλλογισμός – syllogismos – "conclusion," "inference") is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.In its earliest form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced. For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form (without sentence-terminating periods):All men are mortal.Socrates is a man.Therefore, Socrates is mortal.

Modus ponens

In propositional logic, modus ponendo ponens (Latin for "the way that affirms by affirming"; often abbreviated to MP or modus ponens) or implication elimination is a valid, simple argument form and rule of inference. It can be summarized as "P implies Q; P is asserted to be true, so therefore Q must be true." The history of modus ponens goes back to antiquity.

Modus tollens

In propositional logic, modus tollens (or modus tollendo tollens and also denying the consequent) (Latin for "the way that denies by denying") is a valid argument form and a rule of inference.The first to explicitly state the argument form modus tollens were the Stoics.The inference rule modus tollens, also known as the law of contrapositive, validates the inference from P implies Q and the contradictory of Q, to the contradictory of P.

Argument

Testing the Validity of an Argument

Test validity is the extent to which a test (such as a chemical, physical, or scholastic test) accurately measures what it purports to measure. In the fields of psychological testing and educational testing, “Validity refers to the degree to which evidence and theory support the interpretations of test scores entailed by proposed uses of tests.” Although classical models divided the concept into various "validities" (such as content validity, criterion validity, and construct validity), the currently dominant view is that validity is a single unitary construct.

Deductive reasoningDeductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.

PredicateA predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

QuantifierIn logic, quantification is a construct that specifies the quantity of specimens in the domain of discourse that satisfy an open formula. For example, in arithmetic, it allows the expression of the statement that every natural number has a successor. A language element which generates a quantification (such as "every") is called a quantifier. The resulting expression is a quantified expression, it is said to be quantified over the predicate (such as "the natural number x has a successor") whose free variable is bound by the quantifier. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted. Two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. The traditional symbol for the universal quantifier "all" is " ", an inverted letter "∀ A", and for the existential quantifier "exists" is " ", a rotated ∃letter "E". These quantifiers have been generalized beginning with the work of Mostowski and Lindström.Quantification is used as well in natural languages; examples of quantifiers in English are for all, for some, many, few, a lot, and no; see Quantifier (linguistics) for details.

ClauseA clause is a finite disjunction of literals.

Proof by ResolutionIn mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation theorem-proving technique for sentences in propositional logic and first-order logic. In other words, iteratively applying the resolution rule in a suitable way allows for telling whether a propositional formula is satisfiable and for proving that a first-order formula is unsatisfiable. Attempting to prove a satisfiable first-order formula as unsatisfiable may result in a nonterminating computation; this problem doesn't occur in propositional logic.The resolution rule can be traced back to Davis and Putnam (1960); however, their algorithm required to try all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness.The clause produced by a resolution rule is sometimes called a resolvent.

InductionMathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is a form of direct proof, and it is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number. From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers.The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). Mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning and inductive reasoning is excluded from proofs.

CombinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in mathematical optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

Cyclic permutationA cyclic permutation is a permutation in cyclic order of a set of elements.The term circular permutation has the modern meaning of a permutation of objects arranged in a circle (without a distinguished starting object), but in older texts it was used as a synonym for cyclic permutation.

Passwords options numbersHow many passwords can be made of the numbers up to 999999? How many up to 8-symbol passwords can be made with 10 digits (0-9) and 26 English letters (a-z)?

Travelling salesman problemThe travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.TSP is a special case of the travelling purchaser problem.In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (or perhaps exponentially) with the number of cities.The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows may be imposed.

CryptographyCryptography (or cryptology; from Greek κρυπτός kryptós, "hidden, secret"; and γράφειν graphein, "writing", or -λογία -logia, "study", respectively) is the practice and study of techniques for secure communication in the presence of third parties (called adversaries). More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and that are related to various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation. Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce.

Physics

Mid-Term ExamTo prepare to the Mid-Term Exam: Study the class notes, solve all the problem sets, solve all the relevant problems from the textbooks, visit the web-sites, study the solutions, watch the videos, and meet me if you have any questions.

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