Set Complement/Examples/Positive Real Numbers in Complex Numbers

Example of Set Complement

Let the universe $\Bbb U$ be defined to be the set of real numbers $\C$.

Let the set of (strictly) positive real numbers be denoted by $\R_{>0}$.

Then:

$\relcomp {} {\R_{>0} } = \set {x + i y: y \ne 0 \text { or } x \le 0}$

Proof

\(\ds \relcomp {} {\R_{>0} }\)

\(=\)

\(\ds \relcomp {} {\set {x \in \R: x > 0} }\)

\(\ds \)

\(=\)

\(\ds \set {z \in \C: z \notin \set {x \in \R: x > 0} }\)

\(\ds \)

\(=\)

\(\ds \set {x + i y \in C: \neg \paren {y = 0 \text { and } x > 0} }\)

\(\ds \)

\(=\)

\(\ds \set {x + i y \in C: y \ne 0 \text { or } x \le 0 }\)

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Set-Theoretic Notation