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Functional Dependencies
In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two attributes in a relation. Given a relation ''R'' and sets of attributes X,Y \subseteq R, ''X'' is said to functionally determine ''Y'' (written ''X'' → ''Y'') if and only if each ''X'' value in ''R'' is associated with precisely one ''Y'' value in ''R''; ''R'' is then said to ''satisfy'' the functional dependency ''X'' → ''Y''. Equivalently, the projection \Pi_R is a function, i.e. ''Y'' is a function of ''X''. In simple words, if the values for the ''X'' attributes are known (say they are ''x''), then the values for the ''Y'' attributes corresponding to ''x'' can be determined by looking them up in ''any'' tuple of ''R'' containing ''x''. Customarily ''X'' is called the ''determinant'' set and ''Y'' the ''dependent'' set. A functional dependency FD: ''X'' → ''Y'' is called ''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relational Database
A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relational database systems are equipped with the option of using the SQL (Structured Query Language) for querying and maintaining the database. History The term "relational database" was first defined by E. F. Codd at IBM in 1970. Codd introduced the term in his research paper "A Relational Model of Data for Large Shared Data Banks". In this paper and later papers, he defined what he meant by "relational". One well-known definition of what constitutes a relational database system is composed of Codd's 12 rules. However, no commercial implementations of the relational model conform to all of Codd's rules, so the term has gradually come to describe a broader class of database systems, which at a minimum: # Present the data to the user as relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is expressively complete if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lossless Join Decomposition
In database design, a lossless join decomposition is a decomposition of a relation R into relations R_1, R_2 such that a natural join of the two smaller relations yields back the original relation. This is central in removing redundancy safely from databases while preserving the original data. Criteria Lossless join can also be called nonadditive. If R is split into R_1 and R_2, for this decomposition to be lossless (i.e., R_1 \bowtie R_2 = R) then at least one of the two following criteria should be met. Check 1: Verify join explicitly Projecting on R_1 and R_2, and joining them back, results in the relation you started with. Check 2: Via functional dependencies Let R be a relation schema. Let be a set of functional dependencies In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two attributes in a relation. Given a relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Cover
A canonical cover F_c for F (a set of functional dependencies on a relation scheme) is a set of dependencies such that F logically implies all dependencies in F_c, and F_c logically implies all dependencies in F. The set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ... F_c has two important properties: # No functional dependency in F_c contains an extraneous attribute. # Each left side of a functional dependency in F_c is unique. That is, there are no two dependencies a \to b and c \to d in F_c such that a = c. A canonical cover is not unique for a given set of functional dependencies, therefore one set F can have multiple covers F_c. Algorithm for computing a canonical cover # F_c = F # Repeat: ## Use the union rule to replace any dependencies in F_c of the form a \to b and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schema (logic)
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language. The logical form of an argument is called the argument form of the argument. History The importance of the concept of form to logic was already recognized in ancient times. Aristotle, in the '' Prior Analytics'', was probably the first to employ variable letters to represent valid inferences. Therefore, Jan Łukasiewicz claims that the introduction of variables was "one of Aristotle's greatest inventions." According ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is expressively complete if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soundness
In logic or, more precisely, deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system. Definition In deductive reasoning, a sound argument is an argument that is valid and all of its premises are true (and as a consequence its conclusion is true as well). An argument is valid if, assuming its premises are true, the conclusion ''must'' be true. An example of a sound argument is the following well-known syllogism: : ''(premises)'' : All men are mortal. : Socrates is a man. : ''(conclusion)'' : Therefore, Socrates is mortal. Because of the logical necessity of the conclusion, this argument is valid; and because the argument is valid and its premises are true, the argument is sound. However, an argument can be valid without ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inference Rules
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called '' modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armstrong's Axioms
Armstrong's axioms are a set of references (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong in his 1974 paper. The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as F^) when applied to that set (denoted as F). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure F^+. More formally, let \langle R(U), F \rangle denote a relational scheme over the set of attributes U with a set of functional dependencies F. We say that a functional dependency f is logically implied by F, and denote it with F \models f if and only if for every instance r of R that satisfies the functional dependencies in F, r also satisfies f. We denote by F^ the set of all functional dependencies that are logically implied by F. Furthermore, with respect to a set of infe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superkey
In the relational data model a superkey is a set of attributes that uniquely identifies each tuple of a relation. Because superkey values are unique, tuples with the same superkey value must also have the same non-key attribute values. That is, non-key attributes are '' functionally dependent'' on the superkey. The set of all attributes is always a superkey (the ''trivial superkey''). Tuples in a relation are by definition unique, with duplicates removed after each operation, so the set of all attributes is always uniquely valued for every tuple. A '' candidate key'' (or ''minimal superkey'') is a superkey that can't be reduced to a simpler superkey by removing an attribute. For example, in an employee schema with attributes employeeID, name, job, and departmentID, if employeeID values are unique then employeeID combined with any or all of the other attributes can uniquely identify tuples in the table. Each combination, , , , and so on is a superkey. is a candidate key--no subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
