# 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:

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: $123 \ \text{(f)}$