Negative Part of Vertical Section of Function is Vertical Section of Negative Part
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Theorem
Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $x \in X$.
Then:
- $\paren {f_x}^- = \paren {f^-}_x$
where:
- $f_x$ denotes the $x$-vertical function of $f$
- $f^-$ denotes the negative part of $f$.
Proof
Fix $x \in X$.
Then, we have:
\(\ds \map {\paren {f^-}_x} y\) | \(=\) | \(\ds \map {f^-} {x, y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\min \set {0, \map f {x, y} }\) | Definition of Negative Part | |||||||||||
\(\ds \) | \(=\) | \(\ds -\min \set {0, \map {f_x} y}\) | Definition of Vertical Section of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f_x}^-} y\) | Definition of Negative Part |
$\blacksquare$