De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Venn Diagram
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Theorem
- $S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$
Proof
Demonstration by Venn diagram:
The area in blue and magenta is the set difference of $S$ with $T_1$
The area in orange and magenta is set difference of $S$ with $T_2$
The complete shaded area is the set difference of $S$ with the intersection of $T_1$ and $T_2$.
It is also seen to be the union 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): Exercise $1.1: \ 8 \ \text{(h)}$