I'm learning matroids for algorithm analysis. Definition of a matroid is as follows,
In terms of independence, 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:
(I1) The empty set is independent, i.e., ∅ ∈ I.
(I2) Every subset of an independent set is independent, i.e., for each A′⊆ A ⊆ E, if A ∈ I, then A` ∈ I. This is sometimes called the hereditary property, or the downward-closed property.
(I3) If A and B are two independent sets (i.e., each set is independent) and A has more elements than B, then there exists x ∈ A\B such that B ∪ {x} is in I.
This is sometimes called the augmentation property or the independent set exchange property. The first two properties define a combinatorial structure known as an independence system (or abstract simplicial complex).
My problem is if S = {1,2,3,4} and M is a matroid (S,I). Then can we have {1,2,3,4} ∈ I ?
Yes. In general, given a finite set $E$, if we let $2^E$ be the set of all subsets of $E$, then $M=(E,2^E)$ is a matroid, called the free matroid on $E$. Notice that if $M=(E,\mathcal I)$ and $E\in\mathcal I$, then immediately by the hereditary property we must have $\mathcal I=2^E$.