Set Difference with Set Difference is Union of Set Difference with Intersection/Venn Diagram

Theorem

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

Proof

Demonstration by Venn diagram:

Consider the diagram on the left hand side.

The yellow area forms $S \setminus T$.

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

Consider the diagram on the right hand side.

The yellow and orange areas together form $R \cap T$.

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

The red, orange and yellow areas together form $\paren {R \setminus S} \cup \paren {R \cap T}$.

It is seen that the red area on the left hand side is the same as the red, orange and yellow areas together on the right hand side.

$\blacksquare$

Sources

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