Matroids: How can a subset of a set be an independent set?

Question

I am learning about Matroids, but i think i have some misunderstanding in it. The definition of a matroid is as following:

A finite matroid $M$ is a pair $( E , I )$ where $E$ is a finite set (called the ground set) and $I$ is a family of subsets of E (called the independent sets) with the following properties:

1. The empty set is independent, i.e., $\emptyset \in I$.
2. Every subset of an independent set is independent, i.e., for each $A' \subset A \subset E$, if $A \in I$, then $A' \in I$
3. If $A$ and $B$ are two independent sets of $I$ and $A$ has more elements than $B$, there exists $x \in A\backslash B$ such that $B \cup \{x\} \in I$

https://en.wikipedia.org/wiki/Matroid

I could not really find out there what is means with independent sets, so i found that here:

"Two sets A and B are said to be independent if their intersection $A \cap B=\emptyset$.

http://mathworld.wolfram.com/IndependentSet.html

This in combination with statement 2 confuses me. If $A'$ is a subset of $A$, then doesn't it mean that $A \cap A' = A'$?

Again, i am quite sure there is an essential part of this which i am understanding completely wrong, but i just have no idea what it is.

Thanks in advance for any help!

Answer 1

1. There are only finitely many words in the English language, and only a smaller subset that we can easily work with. This means that mathematical English is often forced to use the same words to mean different things in different contexts. The word "normal" comes to mind as a particularly egregious example. In this case, independence means something different in the context of matroids than it does in the context of set theory and/or graph theory and/or probability theory.

2. Matroids seem to generalize the notion of vector spaces. In particular, the notion of an independent set is a generalization of a linearly independent set. You should have that model in your head when you work with matroids. A finite collection of vectors in $\mathbb{R}^2$ is a finite matroid (as long as you throw in the emptyset). An independent set here is a collection of linearly independent vectors, and any linearly independent set has the property that, if you remove a vector, the remaining set is still linearly independent.