Let a * H and b * H be two cosets of H. Consider the following statements: (i)…

Concept. For a subgroup H of a group (G, *), the left cosets of H form a partition of G. The governing criterion is that, for a, b in G, a * H = b * H holds exactly when a-1 * b belongs to H. A partition allows only two relationships between two blocks: empty intersection, or complete coincidence — partial overlap is impossible.

Application to this statement pair.

1. Assume the two cosets are not disjoint, so some element x lies in both a * H and b * H.

2. Membership gives x = a * h1 and x = b * h2 for some h1, h2 in H.

3. Equating the two expressions gives a * h1 = b * h2, hence a-1 * b = h1 * h2-1.

4. H is a subgroup, so it is closed under inverses and under the operation; therefore h1 * h2-1 lies in H, i.e. a-1 * b lies in H.

5. By the equality criterion this forces a * H = b * H, so the two cosets are the same set.

6. So a pair of cosets that shares even one element must coincide; otherwise it shares none. Exactly one of "disjoint" and "identical" therefore holds for every pair, which is the disjunction "(i) or (ii) is true".

Cross-check with a concrete group.

• In (Z, +) with H = 3Z, the cosets 1 + H = {..., 1, 4, 7, ...} and 2 + H = {..., 2, 5, 8, ...} share no element — the disjoint alternative.

• In the same group, 1 + H and 4 + H are the same set, because (-1) + 4 = 3 lies in H — the identical alternative.

• No pair of cosets shares some elements but not all, which is exactly why the cosets of H tile Z without overlap.

Both alternatives genuinely occur and no third possibility exists, so the statement that holds in general is "(i) or (ii) is true".