Dependent Subset of Independent Set Union Singleton Contains Singleton
Jump to navigation
Jump to search
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X$ be an independent subset of $M$.
Let $x \in S$.
Let $C$ be a dependent subset of $M$ such that:
- $C \subseteq X \cup \set x$.
Then:
- $x \in C$
Proof
From the contrapositive statement of Superset of Dependent Set is Dependent:
- $C \nsubseteq X$
From the contrapositive statement of Set Difference with Superset is Empty Set:
- $C \setminus X \ne \O$
From Set Difference over Subset:
- $C \setminus X \subseteq \paren {X \cup \set x} \setminus X = \set x$
From Power Set of Singleton:
- $C \setminus X = \set x$
From Set Difference is Subset:
- $\set x \subseteq C$
From Singleton of Element is Subset:
- $x \in C$
$\blacksquare$