What does this definition of a Primary key mean?
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.