Complement of Horizontal Section of Set is Horizontal Section of Complement

Theorem

Let $X$ and $Y$ be sets.

Let $E \subseteq X \times Y$.

Let $y \in Y$.

Then:

$\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$

where:

$\paren {\paren {X \times Y} \setminus E}^y$ is the $y$-horizontal section of the set difference $\paren {X \times Y} \setminus E$

$E^y$ is the $y$-horizontal section of $E$.

Proof

Note that from the definition of set difference, we have that:

$x \in X \setminus E^y$

if and only if:

$x \in X$ and $x \not \in E^y$.

That is, from the definition of the $y$-horizontal section:

$x \in X$ and $\tuple {x, y} \not \in E$.

This is equivalent to:

$\tuple {x, y} \in \paren {X \times Y} \setminus E$

From the definition of the $y$-horizontal section, this is then equivalent to:

$x \in \paren {\paren {X \times Y} \setminus E}^y$

So we have:

$x \in X \setminus E^y$ if and only if $x \in \paren {\paren {X \times Y} \setminus E}^y$.

So:

$\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$

$\blacksquare$