Horizontal Section of Empty Set
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Theorem
Let $X$ and $Y$ be sets.
Let $y \in Y$.
Then:
- $\O^y = \O$
where $\O$ is the empty set and $\O^y$ is the $y$-horizontal section of $\O$.
Proof
Aiming for a contradiction, suppose suppose that:
- $x \in \O^y$
Then from the definition of the $x$-vertical section, we have:
- $\tuple {x, y} \in \O$
This is impossible from the definition of the empty set.
So:
- there exists no $x \in \O^y$
giving:
- $\O_x = \O$
$\blacksquare$