As Wikipedia defines partition as"A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets(i.e., X is a disjoint union of the subsets)" so for this definition let's take an example as follows:.
Let X={1,2} and let A={(1,1),(2,2),(1,3)} where (1,1)$\in $ A1, (2,2)$\in $ A2 and (1,3)$\in$A3 such that A1, A2, A3 are index subset of A. Here the 1 in (1,1) is different than 1 in (1,3).
Now my question is 'A' a partition of X? Is X a disjoint union of these subsets?