Matroids-Discrete Optimization I happen to stumble across this question where they ask to give an example of an independent system that has two maximal independent sets $X$ and $Y$ such that $|X|\neq |Y|$. How exactly would you formulate such an example? 

An independence system is a collection $C$ of subsets of a finite set $E$ such that if $c \in C$ and $b \subset c$ then $b \in C$. Independence systems are also known as finite (abstract) simplicial complexes. A simplicial complex whose inclusion-maximal subsets all have the same cardinality is called pure. The question asks for a simplicial complex that is not pure. Here is a minimal example. 
Let $E = \{1,2,3\}$ and let $C$ be the set of all subsets of $A = \{1\}$ together with the set of all subsets of $B = \{2,3\}$. So $C = \{\emptyset, \{1\},\{2\},\{3\},\{23\}\}$. Then $C$ is an independence system with (inclusion-)maximal subsets $A$ and $B$. 
For contrast, a matroid is a simplicial complex whose maximal subsets satisfy the matroid basis exchange property. Here's a pure simplicial complex on $E = \{1,2,3,4\}$ that is not a matroid: $C = \{\emptyset, \{1\},\{2\},\{3\},\{4\},\{1,2\},\{3,4\}\}$.
