Set Difference is Right Distributive over Set Intersection/Venn Diagram

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Theorem

$\paren {A \cap B} \setminus C = \paren {A \setminus C} \cap \paren {B \setminus C}$


Proof

Demonstration by Venn diagram:

Set-diff-right-dist-intersection-1.png Set-diff-right-dist-intersection-2.png

Consider the diagram on the left hand side.

The red and yellow areas together form $A \cap B$.

The red area without the yellow area forms $\paren {A \cap B} \setminus C$.


Consider the diagram on the right hand side.

The red and orange areas together form $A \setminus C$.

The yellow and orange areas together form $B \setminus C$.

Their intersection is the orange area, which forms $\paren {A \setminus C} \cap \paren {B \setminus C}$.


It is seen that the red area on the left hand side is the same as the orange area on the right hand side.

$\blacksquare$


Sources