basic understanding of hereditary property in matroid

Question

I'm trying to prove something is a matroid and to do that I must understand what a matroid is. I don't get the hereditary property.

A matroid is an ordered pair $(S, I)$. I is a non-empty family of subsets of $S$ such that if $B \in I$ and $A \subset B$ then $A \in I$

I try understanding this with an example. Let $S=\{2,3,5,9,7\}$ then let $I=\{2,3\}$. For any $A \subset B$ then of course it's going to be in $I$ e.g. $2$ and $3$ are in $B$ and $I$.

I'm assuming I don't understand the definition. Could someone provide a simple example that meats the hereditary property and does not? I don't fully understand the difference between subset ($\subset$) and in ($\in$).

Answer 1

$I$ has to be a set of subsets of $S$, e.g. something like

$$I = \{\{2\}, \emptyset\}.$$

Usually, the sets contained in $I$ are called independent. The hereditary property means that the subset of every independent set again is independent.

You can think of $I$ this way: You have some set $S$ and want to define (however you like) what it means for a subset of $S$ to be independent. You can do this either by giving some formula/condition every subset has to satisfy or you just explicitly list all independent subsets.

Now, some smart minds came to the conclusion that calling a property of sets independence only makes sense if the collection of these sets is a matroid, i.e. if

1. Every subset of an independent set is again independent and
2. if an independent set $A$ is smaller than an independent set $B$, we can add some element of $B$ (which is not already in $A$) to $A$ and get another independent set.