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