Hi I understand the first isomorphism theorem (in context of set theory) intuitively. It lets us turn a surjective image into a bijective one by partitioning them into classes based on the equivalence of the image value.
In my text book this bijection is defined as:
$$ \hat{f}: X/\text{Ker}(f) \rightarrow \operatorname{im}(f) $$
Using my intuitive understanding I see it as an arrow from each partition class to an image value, giving our bijection.
What I do not understand is how my text book defines $\operatorname{Ker}(f)$:
$$ \operatorname{Ker}(f) = \{(x_1,x_2) \in X \times X \mid f(x_1) = f(x_2)\} $$
When I try this on an example I see that it gives me a lot of pairs (including a subset defined by the identity function $\text{id}_A$. I do however not get how that would turn $X/\operatorname{Ker}(f)$ into my desired partition.
What am I missing?
(by the way I know that normally this is seen in the context of Group theory. While we do also see that in my text book later, this theorem is already defined as part of my chapter on set theory).
The comments made by @ArturoMagidin answered my question. I'll summarise it here.
The definition of the set $\text{Ker}(f)$ shows how the equivalence relationship $\sim$ defined on $A$ (for the image $f: A \rightarrow B$).
Once you know how $\sim$ is defined you can form a partition of $A$, identified in the text book an my question as $X/\text{Ker}(f)$. The bijection is as such induced on $f$ using $\sim$ as $\hat{f}: X/\text{Ker}(f) \rightarrow \text{im}(f)$, where $\text{im}(f) \subseteq B$.
The comments of @ArturoMagidin go in more detail.