Minimal hitting set with respect to set inclusion from a book "Parameterized Complexity Theory"
It seems you are confusing the terms minimal and minimum. The Hitting set problem is to find a hitting set of minimum cardinality, not a minimal set. 

A set $S$ is called minimal with respect to some property (in this case, being a hitting set) if there exists no strict subset $T$ of $S$ (i.e. $T\subsetneq S$) that also satisfies that property.

On the other hand, a set $S$ is said to have minimum size or cardinality with respect to some property if there exists no set $T$ of strictly smaller size (i.e. $|T|&lt; |S|$) that also satisfies this property.

A minimum size set must be minimal (why?), but a minimal set need not be of minimum size (why?).

In the example in the book, the set $\{b,c,d\}$ is not a minimal hitting set, because its strict subset $\{c,d\}$ is also a hitting set. The subset $\{b,d,f\}$ is a minimal hitting set, because it has no strict subset that is a hitting set. It is not a minimum hitting set, because there is a hitting set of size $2$ ($\{c,d\}$).