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"The present list of 19 rules of inference constitutes a COMPLETE system of truth-functional logic, in the sense that it permits the construction of a formal proof of validity for ANY valid truth-functional argument." (FN1)
The first nine rules of the list are rules of inference that "correspond to elementary argument forms whose validity is easily established by truth tables." (Id, page 351). The remaining ten rules are the Rules of Replacement, "which permits us to infer from any statement the result of replacing any component of that statement by any other statement logically equivalent to the component replaced." (Id, page 359).
Here are the 19 Rules of Inference:
1. Modus Ponens (M.P.)
p q
p
q 2.Modus Tollens (M.T.)
p q
~q
~p 3.Hypothetical Syllogism (H.S.)p q
q r
p r 4.Disjunctive Syllogism (D.S.)p v q
~ p
q 5. Constructive Dilemma (C.D.)(p q) . (r s)
p v r
q v s 6. Absorption (Abs.)p q
p (p. q)
7. Simplification (Simp.)p . q
p 8. Conjunction (Conj.)p
q
p . q 9. Addition (Add.)p
p v q
Any of the following logically equivalent expressions can replace each other wherever they occur:
10.De Morgan's Theorem (De M.) ~(p . q) (~p v ~q)
~(p v q) (~p . ~q) 11. Commutation (Com.)(p v q) (q v p)
(p . q) (q . p) 12. Association (Assoc.)[p v (q v r)] [(p v q) v r]
[p . (q . r)] [(p . q) . r] 13.Distribution (Dist) [p . (q v r)] [(p . q) v (p . r)]
[p v (q . r)] [(p v q) . (p v r)] 14.Double Negation (D.N.)p ~ ~p 15. Transposition (Trans.)(p q) (~q ~p) 16. Material Implication (M. Imp.)(p q) (~p v q) 17. Material Equivalence (M. Equiv.)(p q) [(p q) . (q p)]
(p q) [(p . q) v (~p . ~q)] 18. Exportation (Exp.)[(p . q) r] [p (q r)] 19. Tautology (Taut.) p (p v p)
p (p . p)
FN1: Introduction to Logic, Irving M. Copi and Carl Cohen, Prentice Hall, Eleventh Edition, 2001, page 361. The book contains the following footnote after this paragraph: "A method of proving this kind of completeness for a set of rules of inference can be found in I. M. Copi, Symbolic Logic, 5th Edition. (New York: Macmillian, 1979), chap 8, See also John A. Winnie, "The Completeness of Copi's System of Natural Deduction," Notre Dame Journal of Formal Logic 11 (July 1970), 379-382."
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Difference between partial dependency and fully functional dependency?
The difference is that partial dependency is when a database's attribute is only partially dependent on the primary key. Fully functional dependency is when the attribute is entirely dependent on the key.
What is Transitive functional dependency?
A functional dependency X->Y is transitive in R, if there exists an attribute Z in R, such that X-> Z, Z-> Y .
What is Partial functional dependency?
Partial Functional Dependency Indicates that if A and B are attributes of a table , B is partially dependent on A if there is some attribute that can be removed from A and yet the dependency still holds. Say for Ex, consider the following functional dependency that exists in the Tbl_Staff table: StaffID,Name -------> BranchID BranchID is functionally dependent on a subset of A (StaffID,Name), namely StaffID. Source :http://www.mahipalreddy.com/dbdesign/dbqa.htm
How functional dependency is related to database table design?
The functional dependency is related to the database table design through the foreign and primary keys. The foreign and primary keys are functionally dependent on each other.
What is functional dependency in DBMS?
A key is a set of attributes that uniquely identifies an entire tuple, a function dependency allow us to express constraints that uniquely identify the values of certain attribute.
Inference rules in DBMS?
Inference rules in database management systems are logical rules used to deduce new information from existing data in databases. They help to generate new facts based on the relationships and constraints defined in the database. Inference rules can be used for data validation, query optimization, and improving data consistency and integrity.
What is non trivial functional dependency?
-->non trivial functional dependency is totally opposite to the trivial functional dependency. --> non trivial dependency means X-->Y that is if Y is not proper subset of X table or relation with X then it said to be non trivial functional dependency.
What is functional dependency in oracle?
In Oracle, functional dependency is when the value of one thing is completely determined by another thing. Functional dependency happens when one attribute determines another attribute in a relation.
Difference between partial dependency and fully functional dependency?
The difference is that partial dependency is when a database's attribute is only partially dependent on the primary key. Fully functional dependency is when the attribute is entirely dependent on the key.
What is Trivial and Non Trivial Dependency?
Trivial functional dependencyA trivial functional dependency is a functional dependency of an attribute on a superset of itself. {Employee ID, Employee Address} → {Employee Address} is trivial, as is {Employee Address} → {Employee Address}.
What is Transitive functional dependency?
A functional dependency X->Y is transitive in R, if there exists an attribute Z in R, such that X-> Z, Z-> Y .
What is difference between Partial Functional Dependency and Full Functional Dependency?
A functional dependency is defined as a constraint between two sets of attributes in a relation from a database.Given a relation R, a set of attributes X in R is said to functionally determine another attribute Y, also in R, (written X→ Y) if and only if eachX value is associated with at most oneY value.A functional dependency X --> Y is full functional dependency if removal of any attribute 'k' from X means that the dependency does not hold any more. Full functional dependency is minimal in size.Partial Functional Dependency Indicates that if A and B are attributes of a table, B is partially dependent on A if there is some attribute that can be removed from A and yet the dependency still holds.A key is a set of attributes that uniquely identifies an entire tuple, a function dependency allow us to express constraints that uniquely identify the values of certain attribute.
Fully functional dependency?
A functional dependency X --> Y is full functional dependency if removal of any attribute 'k' from X means that the dependency does not hold any more.Full functional dependency is minimal in size (contain non-redundant data)In a relation R , attribute B of R is fully functionallydependent on an attribute or set of attributes A of R , if B is functionally dependent on A, but not functionally dependent on any proper subset of A.ORAàB is a fully functionally dependency, if removal of any attribute X from A would result into the cancellation of dependency. i.e. (A-{X})-->B does not hold.
What is Partial functional dependency?
Partial Functional Dependency Indicates that if A and B are attributes of a table , B is partially dependent on A if there is some attribute that can be removed from A and yet the dependency still holds. Say for Ex, consider the following functional dependency that exists in the Tbl_Staff table: StaffID,Name -------> BranchID BranchID is functionally dependent on a subset of A (StaffID,Name), namely StaffID. Source :http://www.mahipalreddy.com/dbdesign/dbqa.htm
How functional dependency is related to database table design?
The functional dependency is related to the database table design through the foreign and primary keys. The foreign and primary keys are functionally dependent on each other.
What is Armstrong inference rules?
Armstrong's Inference RulesSpecify rules for reasoning about dependency functions: Reflexive rule:{Y 1,...,Y n}⊆{X1,...,Xm}implies {X1,...,Xm}→{Y 1,...,Y n} {Name,Sex}→{Name}Augmentation Rule:{X1,...,Xm}→{Y 1,...,Y n}implies {X1,...,Xm,Z}→{Y 1,...,Y n,Z} {Name,Sex}→{Name}implies {Name,Sex,Age}→{Name,Age}Transitive rule:{X1,...,Xm}→{Y 1,...,Y n},{Y 1,...,Y n}→{Z1,...,Zs}imply {X1,...,Xm}→{Z1,...,Zs} {Number}→{Name}{Name}→{Sex}imply{Number}→{Sex}Armstrong inference rules are sound Produce only functional dependencies belonging to the closure complete Produce all the functional dependencies in the closure
How many types of functional dependencies in DBMS?
In DBMS there is three type of functional dependency1 FULL FUNCTION DEPENDENCY.2 PARTIALLY FUNCTION DEPENDENCY.3 TRACIENT FUNCTION DEPENDENCY.Functional Dependency can be classified as follows:Full Functional dependency Indicates that if A and B are attributes(columns)of a table, B is fully functionally dependent on A if B is functionally dependent on A ,but not on any proper subset of A.Partial Functional Dependency Indicates that if Aand B are attributes of a table , B is partially dependent on A if there is some attribute that can be removed from A and yet the dependency still holds.Transitive Functional Dependency: A condition where A , B and C are attributes of a table such that if A is functionally dependent on B and Bis functionally dependent on C then C is Transitively dependent on A via BFor Detailed explanation with example visit Authors original Postinghttp://www.mahipalreddy.com/dbdesign/dbqa.htm
