Set Complement/Examples/Positive Real Numbers in Real Numbers

Example of Set Complement

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

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

Then:

$\relcomp {} {\R_{>0} } = \R_{\le 0}$

the set of non-negative real numbers.

Proof

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

\(=\)

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

\(\ds \)

\(=\)

\(\ds \set {x \in \R: x \not > 0}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \R_{\le 0}\)

$\blacksquare$

Sources

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