# connection between kernel $ker(f)$ and relational kernel $Ker(f)$

What is the connection between the $\text{kernel} \,\,ker(f)$ and the $\text{kernel}\,\,Ker(f)$ of a homomorphism $f$?

I tried finding an explanation of $ker(f)$ vs capitalized $Ker(f)$

I know $ker(f)$ is the set of all elements mapped to the identity

Both $\ker f$ and $\mathrm{Ker} \ f$ are used by mathematicians to refer to the set of elements mapped to the identity. It's just a personal convention as to whether you prefer the notation to be capitalized or not. Same with $\mathrm{im} \ f$ vs. $\mathrm{Im} \ f$ or $\hom(G, H)$ vs. $\mathrm{Hom}(G, H)$ or really any other mathematics notation.