User:Leigh.Samphier/Matroids/Set Difference Then Union Equals Union Then Set Difference/Corollary
< User:Leigh.Samphier | Matroids | Set Difference Then Union Equals Union Then Set Difference(Redirected from User:Leigh.Samphier/Matroids/Corollary of Set Difference Then Union Equals Union Then Set Difference)
Jump to navigation
Jump to search
This page 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. |
Corollary of Set Difference Then Union Equals Union Then Set Difference/Corollary
Let $S, T$ be sets.
Let $A \subseteq S \setminus T$.
Let $B \subseteq T \setminus S$.
Then:
- $\paren{S \setminus A} \cup B = \paren{S \cup B} \setminus A$
Proof
From Set Difference is Disjoint with Reverse:
- $\paren{S \setminus T} \cap \paren{T \setminus S} = \O$
From Subsets of Disjoint Sets are Disjoint:
- $A \cap B = \O$
From User:Leigh.Samphier/Matroids/Set Difference Then Union Equals Union Then Set Difference:
- $\paren{S \setminus A} \cup B = \paren{S \cup B} \setminus A$
$\blacksquare$