User:Leigh.Samphier/Matroids/Set Difference Then Union Equals Union Then Set Difference/Corollary

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:

$\paren{S \setminus A} \cup B = \paren{S \cup B} \setminus A$

$\blacksquare$