1. Functional dependencies rule out certain tuples from appearing in a relation.

If A →B, then we cannot have two tuples with the same A value but different B values.

2. Multivalued dependencies do not rule out the existence of certain tuples.

Instead, they require that other tuples of a certain form be present in the relation.

3. Let R be a relation schema, and let α ∈ R and β ∈ R.

The multivalued dependency α→→ β

Figure 7.5: Tabular representation of α→→ β.

Figure 7.6: (name; address; car) where name →→address and name →→car.

holds on R if in any legal relation r(R), for all pairs of tuples t1 and t2 in r such that t1[ α] = t2[ α], there exist tuples t3 and t4 in r such that:

t1[ α] = t2[ α] = t3[ α] = t4[ α]
t3[ β] = t1[ β]
t3[R- β] = t2[R- β]
t4[ β ] = t2[ β ]
t4[R- βí°€ ] = t1[R- βí°€ ]

4. Figure 7.5 (textbook 6.10) shows a tabular representation of this. It looks horrendously complicated, but is
really rather simple. A simple example is a table with the schema (name; address; car), as shown in Figure
7.6.

• Intuitively, α→→ βsays that the relationship between and is independent of the relationship between and R.ô€€€
• If the multivalued dependency α→→ βis satisfied by all relations on schema R, then we say it is a trivial multivalued dependency on schema R.
• Thus α→→ βis trivial β∈α if or β∪ α= R.

5. Look at the example relation bc relation in Figure 7.7 (textbook 6.11).

• We must repeat the loan number once for each address a customer has.
• We must repeat the address once for each loan the customer has.
• This repetition is pointless, as the relationship between a customer and a loan is independent of the relationship between a customer and his or her address.

Figure 7.7: Relation bc, an example of redundancy in a BCNF relation.

Figure 7.8: An illegal bc relation.

• If a customer, say "Smith", has loan number 23, we want all of Smith's addresses to be associated with that loan.
• Thus the relation of Figure 7.8 (textbook 6.12) is illegal.
• If we look at our de nition of multivalued dependency, we see that we want the multivalued dependency cname →→street ccity to hold on BC-schema.

6. Note that if a relation r fails to satisfy a given multivalued dependency, we can construct a relation r0 that does satisfy the multivalued dependency by adding tuples to r.