What does this definition of a Primary key mean? My text book gives the following definition of a primary key in a relational database, which I don't entirely understand. Help would be greatly appreciated. 


  Let $R$ be a relation. Then the primary key for $R$ is a subset of the
  set of attributes of $R$, say $K$, satisfying the following two properties:
  
  
  Uniqueness Property: No two distinct tuples of $R$ have the same value for $K$. 
  Irreducibility Property: No proper subset of $K$ has the uniqueness property.
  


I'm getting lost by the Irreducibility property.
Note how $K$ can be a set of columns. Irreducibility means that you have to pick minimal sets of columns.

Nota bene: They should require $K \neq \emptyset$.

For instance, consider this relation.

A   B   C

1   4   4
2   4   6
3   6   6


Let us investigate all possible keys.


A -- unique and irreducible.
B -- not unique.
C -- not unique.
A,B -- reducible to A.
A,C -- reducible to A.
B,C -- unique and irreducible.
A,B,C -- reducible to A.


Hence, there are two choices for primare keys here: A and B,C.