Set Theory/Examples/Union of Intersections with Set Complements/3 Circles in Complex Plane

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Example in Set Theory

Let $A$, $B$ and $C$ be sets defined by circles embedded in the complex plane as follows:

\(\displaystyle A\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + i} < 3}\)
\(\displaystyle B\) \(=\) \(\displaystyle \set {z \in \C: \cmod z < 5}\)
\(\displaystyle C\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + 1} < 4}\)


$\paren {A \cap \tilde B} \cup \paren {B \cap \tilde C} \cup \paren {C \cap \tilde A}$, where $\tilde A$ denotes the complement of $A$, can be illustrated graphically as:

Set-Union-of-Intersection-Complements-3-Circles-in-Complex-Plane.png

where the union of the intersections is depicted in yellow.


Sources