Set Difference with Union/Venn Diagram

Theorem

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

Proof

Demonstration by Venn diagram:

Consider the diagram on the left hand side.

The red area forms $R \setminus \paren {S \cup T}$.

Consider the diagram in the middle.

The red and orange areas together form $R \setminus S$.

The red area alone forms $\paren {R \setminus S} \setminus T$.

Consider the diagram on the right hand side.

The red and orange areas together form $R \setminus T$.

The red area alone forms $\paren {R \setminus T} \setminus S$.

It is seen that the red areas are the same on all diagrams.

$\blacksquare$

Sources

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