Set Difference with Set Difference is Union of Set Difference with Intersection

Theorem

Let $R, S, T$ be sets.

Then:

$R \setminus \paren {S \setminus T} = \paren {R \setminus S} \cup \paren {R \cap T}$

where:

$S \setminus T$ denotes set difference

$S \cup T$ denotes set union

$S \cap T$ denotes set intersection.

Illustration by Venn Diagram

Corollary

$T \setminus \paren {S \setminus T} = T$

Proof

Consider $R, S, T \subseteq \mathbb U$, where $\mathbb U$ is considered as the universe.

\(\ds R \setminus \paren {S \setminus T}\)

\(=\)

\(\ds R \cap \overline {\paren {S \cap \overline T} }\)

Set Difference as Intersection with Complement

\(\ds \)

\(=\)

\(\ds R \cap \paren {\overline S \cup \overline {\paren {\overline T} } }\)

De Morgan's Laws

\(\ds \)

\(=\)

\(\ds R \cap \paren {\overline S \cup T}\)

Complement of Complement

\(\ds \)

\(=\)

\(\ds \paren {R \cap \overline S} \cup \paren {R \cap T}\)

Intersection Distributes over Union

\(\ds \)

\(=\)

\(\ds \paren {R \setminus S} \cup \paren {R \cap T}\)

Set Difference as Intersection with Complement

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(d)}$