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- 1. DefinitionA functional dependency is defined as aconstraint between two sets of attributesin a relation from a database.Given a relation R, a set of attributes X in Ris said to functionally determine anotherattribute Y, also in R, (written X → Y) ifand only if each X value is associatedwith at most one Y value.
- 2. In other words….X is the determinant set and Y is thedependent attribute. Thus, given atuple and the values of the attributes inX, one can determine thecorresponding value of the Y attribute.
- 3. ExampleEmployeeSSN Name JobType DeptName557-78-6587 Lance Smith Accountant Salary214-45-2398 Lance Smith Engineer ProductNote: Name is functionally dependent on SSN because an employee’s namecan be uniquely determined from their SSN. Name does not determine SSN,because more than one employee can have the same name..
- 4. KeysWhereas a key is a set of attributes that uniquelyidentifies an entire tuple, a functionaldependency allows us to express constraintsthat uniquely identify the values of certainattributes.However, a candidate key is always adeterminant, but a determinant doesn’t needto be a key.
- 5. ClosureLet a relation R have some functional dependencies Fspecified. The closure of F (usually written as F+) isthe set of all functional dependencies that may belogically derived from F. Often F is the set of mostobvious and important functional dependencies andF+, the closure, is the set of all the functionaldependencies including F and those that can bededuced from F. The closure is important and may,for example, be needed in finding one or morecandidate keys of the relation.
- 6. ExampleStudentSNo SName CNo CName Addr Instr. Office5425 SusanRoss102 Calc I …SanJose,CAP. Smith B42Room1127845 DaveTurco541 Bio 10 ...SanDiego,CAL. Talip B24Room210SNo -> SName CNo -> CName Instr -> OfficeSNo -> Addr CNo -> Instr
- 7. AxiomsBefore we can determine the closure ofthe relation, Student, we need a set ofrules.Developed by Armstrong in 1974, thereare six rules (axioms) that all possiblefunctional dependencies may bederived from them.
- 8. Axioms Cont.1. Reflexivity Rule --- If X is a set of attributes andY is a subset of X, then X Y holds.each subset of X is functionally dependent on X.2. Augmentation Rule --- If X Y holds and W isa set of attributes, then WX WY holds.3. Transitivity Rule --- If X Y and Y Z holds,then X Z holds.
- 9. Derived Theorems fromAxioms4. Union Rule --- If X Y and X Zholds, then X YZ holds.5. Decomposition Rule --- If X YZholds, then so do X Y and X Z.6. Pseudotransitivity Rule --- If X Yand WY Z hold then so does WX Z.
- 10. Back to the ExampleSNo SName CNo CName Addr Instr. OfficeBased on the rules provided, the following dependencies can bederived.(SNo, CNo) SNo (Rule 1) -- subset(SNo, CNo) CNo (Rule 1)(SNo, CNo) (SName, CName) (Rule 2) -- augmentationCNo office (Rule 3) -- transitivitySNo (SName, address) (Union Rule)etc.
- 11. Too Many FDsUsing the first rule alone, from our example wehave 2^7 = 128 subsets. This will further leadto many more functional dependencies. Thisdefeats the purpose of normalizing relations.So what now?One way is to deal with one attribute or a set ofattributes at a time and find its closure (i.e. allfunctional dependencies relating to them). Theaim of this exercise is to find what attributesdepend on a given set of attributes andtherefore ought to be together.
- 12. The approachStep 1 Let X^c <- XStep 2 Let the next dependency be A -> B.If A isin X^c and B is not, X^c <- X^c + B.Step 3 Continue step 2 until no newattributes canbe added to X^c.
- 13. Back To Our ExampleConsider the following relation: student(SNo, SName, CNo,CName).We wish to determine the closure of (SNo, CNo). We have thefollowing functional dependencies.SNo -> SNameCNo -> CName Step 1 --- X^c <- X, that is, X^c <- (SNo, CNo)Step 2 --- Consider SNo -> SName, since SNo is in X^c and SName is not, wehave: X^c <- (SNo, CNo) + SNameStep 3 --- Consider CNo -> CName, since CNo is in X^c and CName is not, wehave: X^c <- (SNo, CNo, SName) + CNameStep 4 --- Again, consider SNo -> SName but this does not change X^c.Step 5 --- Again, consider CNo -> CName but this does not change X^c. Therefore X+ = X^c = (SNo, CNo, SName, CName). This shows that all the attributes in the relation student (SNo, CNo, SName,CName) are dependent on (SNo, CNo) and therefore (SNo, CNo) is a candidate keyof the present relation. In this case, it is the only candidate key.
- 14. Normal Form Initially Codd (1972) presented three normalforms (1NF, 2NF and 3NF) all based onfunctional dependencies among the attributesof a relation. Later Boyce and Codd proposedanother normal form called the Boyce-Coddnormal form (BCNF). The fourth and fifthnormal forms are based on multi-value andjoin dependencies and were proposed later. The primary objective of normalization is toavoid anomalies.

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