Formal defintion of SET-PARTITION as a language I am not quite sure howto define SET-PARTITION as a language as in Sipser. Is it

$$
\left\{ \langle S,A,B\rangle  \;\middle|\; (A,B) \text{ is partition of } S \text{ and } \sum_{n\in A} n = \sum_{n\in B} n \right\}
\quad \text{ or } \\
\left\{ \langle S\rangle   \;\middle|\; \text{ there exists a partition } (A,B) \text{ of } S \text{ with } \sum_{n\in A} n = \sum_{n\in B} n\right\} \quad ?
$$
The set partition problem asks whether there exists a 2-partition of a multi-set such that sum of numbers in one set is equal to the sum of numbers in the other set.

Note the phrase whether there exists; if you are already being given the partition in the input $\langle S,A,B \rangle$ then the problem becomes whether the given partition is valid and sum of elements of $A$ and $B$ are equal. This problem is easy and can be solved in polynomial it time. 

However, Set Partition is NP-Complete, which seems unlikely to be solved in polynomial time.