So I noticed a rather interesting thing about the image function and preimage function. The wikipedia page for Image: https://en.wikipedia.org/wiki/Image_(mathematics) (notation for image section)
The construction they are describing there can be done for general binary relations $R$, not just for functions. Interesingly the result structures from image and preimage are still functions. Using the arrow notation from the article, we then have that:
Given a binary relation $R$ with domain $X$ and codomain $Y$. There exist functions:
$R^{\rightarrow}:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)$
and
$R^{\leftarrow}:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)$
The wikipedia article makes no mention of this, nor is it discussed in their article on binary relations. I'm kinda curious how much is known about this (and it's also worth pointing out that you can iterate this construction and talk about the image map of the image map etc; arrow notation is really useful for this.). I noticed that some of the properties that the image maps have for functions do not hold for general binary relations (e.g. injective function yields injective image map, but that is no longer true for injective binary relations).