On $\Bbb Z_3$, we typically define addition and multiplication as follows:
$$[a]+[b]=[a+b]$$ $$ [a]\cdot[b]=[a\cdot b]$$
Consider defining addition as $[a]+[b]=[0]$ for all $a,b\in \Bbb Z$. This addition is well defined:
Let $a\equiv c \pmod 3$ and $b\equiv d \pmod 3$ for $a,b,c,d \in \Bbb Z$. Then $[a]=[c]$ and $[b]=[d]$, which implies: $$[a]+[b]=[c]+[d]=[0]$$ However, $f([a],[b])=[a]+[b]$ is not an onto mapping from $\Bbb Z_3 \times \Bbb Z_3$ to $\Bbb Z_3$.
EDIT (clarification of question): Is the above definition for addition valid?
That is, is it possible to define addition on $\Bbb Z_3$ in such a way that the function $f:\Bbb Z_3 \times \Bbb Z_3 \rightarrow \Bbb Z_3$ such that $f([a],[b])=[a]+[b]$ is not onto? And likewise for multiplication?
Thank you for your help. I appreciate an explanation suitable for an undergraduate like myself.
"Addition" operations are usually supposed to have an additive identity - let's call it z. In particular, we should have [1]+z=[1]. What is your $z$?
As to your more general question, the very standard expectations of an additive identity and additive inverses always imply that addition is onto.
What you're doing is abusing nomenclature. You can call your shoes "a car", and while they are a well defined method of transportation and your 4-year old might enjoy it, you can't sit in them and they don't have a steering wheel or gas or brake pedal. You shouldn't really be calling them "a car".
(And the situation for multiplication is similar - assuming you want to define a field and not just a ring. If you set the zero element aside, the expectation is that a multiplicative identity exists, and multiplicative inverses, so multiplication is also onto.
It's hard to tell what level you're at only knowing that you're an undergrad, but these questions are answerable in ones first abstract algebra course, and follow pretty quickly from the group and field axioms.)