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

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 $A \cap B=\emptyset$.

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!

• You are confusing two different notions of independent set. I would advise forgetting about the Wolfram reference... – Lord Shark the Unknown Nov 26 '17 at 14:35
• The three properties that you give are the definition of an independent set in this theory. It is a different definition than those you cite. A closer analog here would be a linearly independent set. – Xander Henderson Nov 26 '17 at 14:46

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.
• @user3053216 That is what I mean. Independence in a matroid is meant to generalize the notion of a linearly independent set in a vector space. However, note that in vector-land, a set $\{ v_j \}$ is linearly independent if $\sum_j a_j v_j = 0$ if and only if $a_j = 0$ for all $j$. In a matroid, we may not have addition or scalar multiplication, so we need a slightly different definition that still captures the important relations. Hence the definition you give above. – Xander Henderson Nov 26 '17 at 15:40