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# CHAPTER 4

4.1 The complete set of FDs—what’s known, formally, as the closure, though it has nothing to do with the closure property of the relational algebra—for relvar SP contains a total of 31 FDs. Here they are:

```{ SNO , PNO , QTY } → { SNO , PNO , QTY }
{ SNO , PNO , QTY } → { SNO , PNO }
{ SNO , PNO , QTY } → { SNO , QTY }
{ SNO , PNO , QTY } → { PNO , QTY }
{ SNO , PNO , QTY } → { SNO }
{ SNO , PNO , QTY } → { PNO }
{ SNO , PNO , QTY } → { QTY }
{ SNO , PNO , QTY } → { }

{ SNO , PNO }       → { SNO , PNO , QTY }
{ SNO , PNO }       → { SNO , PNO }
{ SNO , PNO }       → { SNO , QTY }
{ SNO , PNO }       → { PNO , QTY }
{ SNO , PNO }       → { SNO }
{ SNO , PNO }       → { PNO }
{ SNO , PNO }       → { QTY }
{ SNO , PNO }       → { }

{ SNO , QTY }       → { SNO , QTY }
{ SNO , QTY }       → { SNO }
{ SNO , QTY }       → { QTY }
{ SNO , QTY }       → { }

{ PNO , QTY }       → { PNO , QTY }
{ PNO , QTY }       → { PNO }
{ PNO , QTY }       → { QTY }
{ PNO , QTY }       → { }

{ SNO }             → { SNO }
{ SNO }             → { }

{ PNO }             → { PNO }
{ PNO }             → { }

{ QTY }             → { QTY }
{ QTY }             → { }

{ }                 → { }```

The only ones that aren’t trivial are the following four:

```{ SNO , PNO } → { SNO , PNO , QTY }
{ SNO , PNO } → { SNO , QTY }
{ SNO , PNO } → { PNO , QTY }
{ SNO , PNO } → { QTY }```

The only irreducible ones are the following eleven:

```{ SNO , PNO } → { SNO , PNO , QTY }
{ SNO , PNO } → { SNO , PNO }
{ SNO , PNO } → { SNO , QTY }
{ SNO , PNO } → { PNO , QTY }
{ SNO , PNO } → { QTY }

{ SNO , QTY } → { SNO , QTY }
{ PNO , QTY } → { PNO , QTY }
{ SNO }       → { SNO }
{ PNO }       → { PNO }
{ QTY }       → { QTY }
{ }           → { }```

4.2 ...

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