Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = x y

Example of Preimage of Subset under Mapping

Yellow: $f^{-1} \sqbrk S$, orange: $f^{-1} \sqbrk T$ (boundaries not included)

Let $f: \R^2 \to \R$ be the real function of $2$ variables defined as:

$\forall \tuple {x, y} \in \R^2: \map f {x, y} = x y$

Let the following subsets of $\R^2$ be defined:

\(\ds S\)

\(=\)

\(\ds f^{-1} \sqbrk {\openint 1 \to}\)

\(\ds T\)

\(=\)

\(\ds f^{-1} \sqbrk {\openint 0 1}\)

Then:

\(\ds S\)

\(=\)

\(\ds \set {\tuple {x, y} \in \R^2: x y > 1}\)

\(\ds T\)

\(=\)

\(\ds \set {\tuple {x, y} \in \R^2: 0 < x y < 1}\)

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.16 \ \text {(a)}$