De Morgan's Laws (Set Theory)/Set Difference/Difference with Union/Venn Diagram

Theorem

$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$

Proof

Demonstration by Venn diagram:

The area in orange and red is the set difference of $S$ with $T_1$

The area in orange and yellow is set difference of $S$ with $T_2$

The area in orange only is the set difference of $S$ with the union of $T_1$ and $T_2$.

It is also seen to be the intersection of the set difference of $S$ with $T_1$ and the set difference of $S$ with $T_2$.

$\blacksquare$

Source of Name

This entry was named for Augustus De Morgan.

Sources

• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.8$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.6$. Difference and complement: Example $21$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(a)}$