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 example can be constructed by matchings in a graph, where the independent sets are the edges of a matching. Here is a particular case:

On the left is the graph and next to it, marked in red, the edges of two maximal independent sets with different cardinality. 
If we call the edges from left to right 1,2,3. Then the set of all matchings is $\lbrace \emptyset, 1, 2, 3, 13\rbrace$, where $\lbrace 2\rbrace$ and $\lbrace 1,3\rbrace$ denote the maximal independent sets.
