ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4765 on D5612: Euclidean real set

Reflection preserves euclidean real subset relation

Formulation 0

Let $E, F \subseteq \mathbb{R}^N$ each be a D5612: Euclidean real set such that

(i)	\begin{equation} E \subseteq F \end{equation}

Then \begin{equation} - E \subseteq - F \end{equation}

Proofs

Proof 0

Let $E, F \subseteq \mathbb{R}^N$ each be a D5612: Euclidean real set such that

(i)	\begin{equation} E \subseteq F \end{equation}

If $- E$ is empty, then the claim is a consequence of result R7: Empty set is subset of every set, so assume that $- E \neq \emptyset$. If $x \in - E$, then $x = - e$ for some $e \in E$. Since $E \subseteq F$, then $e \in F$ and thus $x = - e \in - F$. Since $x \in - E$ was arbitrary, we are done. $\square$