Problem Axiom of addition If X Y and

Problem • Axiom of addition: – If X Y and W V then XW YV • Prove this axiom using Armstrong’s Axioms or show that it is not sound 1

Problem • Axiom of confusion: – If XZ Y, YZU X, W V, V WU then XW YV • Prove this axiom using Armstrong’s Axioms or show that it is not sound 2

Problem • R = (A, B, C, D). For each of the following: – find all keys – determine the normal form of R 1. C → D, C → A, B → C 2. ABC → D, D → A 3. A → B, BC → D, A → C 4. AB → C, AB → D, C → A, D → B 3

Problem • Proposition: Let R be a relation containing exactly two attributes. Let F be a set of functional dependencies. Then, R is in BCNF. • Prove or give a counter-example. 4

Problem • Proposition: Let R be a relation and F be a set of functional dependencies. Suppose that every key in F has a single attribute. Then, R is in BCNF. • Prove or give a counter-example. 5

Problem • Proposition: Let R be a relation and F be a set of functional dependencies. Suppose that every key in F has a single attribute. Then, R is in BCNF if and only if R is in 3 NF. • Prove or give a counter-example. 6

Problem • Proposition: Let R be a relation and F be a set of functional dependencies. Suppose that every dependency in F has one attribute on the left side. Then, R is in BCNF if and only if R is in 3 NF • Prove or show a simple counter example. 7

Problem • Find small(est) examples of R and F such that: – R is in BCNF – R is in 3 NF, but not in BCNF – R is not in 3 NF 8

Problem • R=(ABCD) • F={A->B, C->D, A->D} • What is the normal form of R? • We have a decomposition R 1=(ABC), R 2=(BCD) • Is this a lossless join decomposition? 9

Problem • R=(ABC) • F={A->B, BC->A} • What is the normal form of R? • We have a decomposition R 1=(AB), R 2=(AC) • Is this a lossless join decomposition? 10

Problem • Let R=ABCDEG • Let F={C->D, E->C, EG->A, G->B} • What are all the keys of R? • What is the normal form of R? • Is the following decomposition lossless? – R 1=AEG, R 2=CE, R 3=BG, R 4=DEG 11

Problem • Prove or give a counter example – (X+)+ = X+ – (X Y)+ = X + Y+ 12

Problem • Consider the following axiom system: (B 1) X X (B 2) X YZ then X Y, X Z (B 3) X YZ, Z C then X YZC • Prove that this system is complete by showing how each of Armstrong’s Axioms can be proven from the axioms above 13