Union of Intersections/Examples/4 Arbitrarily Chosen Sets of Complex Numbers
Jump to navigation
Jump to search
Example of Union of Intersections
Let:
\(\ds A\) | \(=\) | \(\ds \set {1, i, -i}\) | ||||||||||||
\(\ds B\) | \(=\) | \(\ds \set {2, 1, -i}\) | ||||||||||||
\(\ds C\) | \(=\) | \(\ds \set {i, -1, 1 + i}\) | ||||||||||||
\(\ds D\) | \(=\) | \(\ds \set {0, -i, 1}\) |
Then:
- $\paren {A \cap C} \cup \paren {B \cap D} = \set {1, i, -i}$
Proof
\(\ds \paren {A \cap C} \cup \paren {B \cap D}\) | \(=\) | \(\ds \paren {\set {1, i, -i} \cap \set {i, -1, 1 + i} } \cup \paren {\set {2, 1, -i} \cap \set {0, -i, 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set i \cup \set {-i, 1}\) | Definition of Set Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {1, i, -i}\) | Definition of Set Union |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Point Sets: $121 \ \text{(b)}$