Michael, the usual sense in which an equivalence relation is determined by a function is by identifying any elements which map to the same value. (The partition is then the set of equivalence classes.) This only works for partitioning the domain relative to the codomain, so \$$0 \to S\$$ would have to generate a partition on \$$0\$$, not on \$$S\$$. We could consider a different construction, in which elements of S are identified if they're mapped to by the same input, but this gives a discrete partition for _all_ functions, not just \$$0 \to S\$$. This is because for a function to be a function, an input can only map to one output; therefore no two values in the codomain can be mapped to by a single value in the domain. Relative to this alternative construction, \$$0 \to S\$$ is not particularly special.