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

Example in Set Theory

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

\(\ds A\)

\(=\)

\(\ds \set {z \in \C: \cmod {z + i} < 3}\)

\(\ds B\)

\(=\)

\(\ds \set {z \in \C: \cmod z < 5}\)

\(\ds C\)

\(=\)

\(\ds \set {z \in \C: \cmod {z + 1} < 4}\)

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

where the union of the intersections is depicted in yellow.

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Point Sets: $123 \ \text{(f)}$