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

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Theorem

Let $S, A, B$ be sets.

Let $A \subseteq S$.

Let $A \cap B = \O$.

Then:

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

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

We have:

\(\ds \paren{S \cup B} \setminus A\)

\(=\)

\(\ds \paren {S \setminus A} \cup \paren {B \setminus A}\)

Set Difference is Right Distributive over Union

\(\ds \)

\(=\)

\(\ds \paren {S \setminus A} \cup B\)

Set Difference with Disjoint Set

$\blacksquare$