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Functional Dependence

Functional Dependence. Lecture 11 Inst: Haya Sammaneh. Functional Dependence. Existence dependence: The existence of B depends on A Functional dependence : B’s value depends on A’s value EmpName is functionally dependent on EmpNo

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Functional Dependence

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  1. Functional Dependence Lecture 11 Inst: Haya Sammaneh

  2. Functional Dependence • Existence dependence: The existence of B depends on A • Functional dependence: B’s value depends on A’s value • EmpName is functionally dependent on EmpNo • Given the EmpNo, I can one and only one value of EmpName • Require that the value for a certain set of attributes determines uniquely the value for another set of attributes. • Functional dependence is a generalization of the notion of a key.

  3. Functional Dependencies • Loan-info = (branch-name, loan-number, customer-name, amount)We expect the following set of functional dependencies to hold: loan-number  amount loan-number  branch-namebut would not expect the following to hold: loan-number  customer-name

  4. Examples loan-number  amountloan-number  branch-nameloan-number  customer-name  Another example: reverse of the fd’s above

  5. Closure of a Set of Functional Dependencies • Given a set of functional dependencies F, there are certain other functional dependencies that are logically implied by F. • The set of all functional dependencies logically implied by F is the closure of F. • We denote the closure of F by F+. • We can find all of F+ by applying : • if   , then    (reflexivity) • if   , then    (augmentation) زياده • if    and  , then    (transitivity)تعدي

  6. Closure • We can further simplify computation of F+ by using the following additional rules. • If    holds and    holds, then    holds (union) • If    holds, then    holds and    holds (decomposition) • If    holds and    holds, then    holds (pseudotransitivity)

  7. Examples of Armstrong’s Axioms • We can find all of F+ by applying : • if   , then    (reflexivity)loan-no  loan-no loan-no, amount  loan-noloan-no, amount  amount • if   , then    (augmentation)loan-no  amount (given)loan-no, branch-name amount, branch-name • if    and  , then    (transitivity)loan-no  branch-name (given) branch-name  branch-city (given)loan-no  branch-city

  8. Example • R = (A, B, C, G, H, I) • F = {A  B A  C CG  H • CG  I • B  H} • some members of F+ • A  H • AG  I • CG  HI A  B; B  H A  C; AG  CG; CG  I

  9. result contains all of the attributes of R, so stop Example • R = (A, B, C, G, H, I)F = ( A  B A  C CG  H CG  I B  H} • (AG+)1. Result= AG2. Result= ABCG (A  C; A  B and A  AG)3. Result= ABCGH (CG  H and CG  AGBC)4. Result=ABCGHI (CG  I and CG  AGBCH) • Is AG a candidate key?1. AG  R

  10. Example • R = (A, B, C, G, H, I)F = { A BA CCG HCG IB H} • some members of F + • A H • by transitivity from A B and B H • AG I • by augmenting A C with G, to get AG CG and then transitivity with CG I • CG HI • from CG H and CG I : “union rule”

  11. Example Given this FD for this R(A,B,C,D,E,F) AB C AD E BD AFB Check if AB+ is a key for this relation? AB+ is key if AB+ can find all the attribute of R ABAB BD so B AB  AB+ABD ADE so AD ABDAB+ABDE ABC so AB ABDE AB+ABCDE AFB so AF Not ABDE  AB+ABCDE AB not a key because it does not contain all attributes such as F

  12. Example Given this FD for this R(A,B,C,D,E,F) AB C AD E BD AFB Check if AF+ is a key for this relation? AF is a key

  13. Example Given this FD for this R(A,B,C,D,E,F,G) A D D CG BE EF ABF What are all the keys for this relation? Solution: we look to the right side of FD and take all the attribute which does not found in the FD Here is AB Then find AB+ = ABCDEFG  AB is a key

  14. Example Given this FD for this R(A,B,C,D) AB C CD DA What are all the keys for this relation? Look to the right  B Take all minimum combination with B AB,BC,BD Find AB+=ABCD BC+=ABCD BD+=ABCD ALL AB,BC,BD are keys List all the super keys for R that are not key (not minimal number of attributes) ABC BCD ABCD

  15. Trivial and non trivial dependency • A Functional dependency A1,A2,….An Bn is said to be trivial dependency if B is one of A’s such as: • title, year  title • Not trivial if one of B’s not on A’s such as: • title, year length, year • Complete not trivial dependency if all B’s not found on A’s

  16. Example Given this FD for this R(A,B,C,D) AC D  completely non trivial BC A  completely non trivial DB  completely non trivial What are the keys? Look to right  C  take all combinations  AC, BC, CD What are the super key? ABC ADC BCD ABCD ABD not a super key because C must be found on left side

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