Lets say we have a relation A x B. As far as I unterstood, in a right-unique relation, for every element from A, there is at least one element in B. But there might be elements in B which do not have a "partner" in A.
My understanding of left-unique is, that every element from B has only one partner in A.
So, when a Relation is left-unique, it has to be right-unique. Is this correct, or did I messed something up?
On
You're confusing unique with a serial relation. However for this the property still doesn't hold. Take A = B = {0,1} and R = {<0, 0>, <1, 0>}; i.e., xRy if and only if y = 0. This relation is serial in x but not in y.
A right-unique relation between $A$ and $B$ means $\forall a \in A, a$ is related to at most one $b \in B$.
A left-unique relation between $A$ and $B$ means $\forall b \in B$, at most one $a \in A$ is related to $b$.
Example of a left-unique relation that is not right-unique:
$$A = \{1,2,3\}, \quad B = \{4,5,6\}$$ $$R = \{(1,4),\,(1,5)\}.$$