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

 $\displaystyle \relcomp {} {\R_{>0} }$ $=$ $\displaystyle \relcomp {} {\set {x \in \R: x > 0} }$ $\displaystyle$ $=$ $\displaystyle \set {z \in \C: z \notin \set {x \in \R: x > 0} }$ $\displaystyle$ $=$ $\displaystyle \set {x + i y \in C: \neg \paren {y = 0 \text { and } x > 0} }$ $\displaystyle$ $=$ $\displaystyle \set {x + i y \in C: y \ne 0 \text { or } x \le 0 }$

$\blacksquare$