I have a question with regards to surjectivity and multivalued functions. If I have a domain with one element and a mapping to the codomain, in which the codomain consists of three elements and that element in the domain is mapped to all the three distinct elements in the codomain, then is the mapping surjective? Because for all elements in the codomain there is one in the domain which maps to it. But I haven't seen that as an example of surjectivity, may someone explain to me why? or is it because I just haven't found an example yet?
It's not a function that you defined but a relation which is total on the codomain.
A relation (or multivalued partial function, if you prefer) $A\to B$ is a function, by definition, iff it is single-valued and total on the domain, meaning that every $a\in A$ is in relation with a unique $b\in B$.
The term 'surjective' is usually applied only to functions.