From the following instance of a relation schema R(A, B, C), we can conclude…

Concept. A functional dependency X → Y means: any two tuples that agree on X must also agree on Y. The schema names the attributes and constraints; the four displayed rows are only one relation instance. A single instance can DISPROVE a dependency — one pair of tuples that agree on X but differ on Y is a decisive counterexample — but it cannot PROVE a dependency for the schema, because another legal tuple might violate it. Therefore, an instance supports a certain schema-level conclusion only when it supplies a counterexample that refutes a dependency.

Apply it to this instance.

1. Test B → C: the two tuples with B = 1 have C = 1 and C = 0. They agree on B but differ on C — a counterexample. Hence B → C is refuted and 'B does not functionally determine C' is a certain conclusion.

2. Test A → B: A = 1 always pairs with B = 1, and A = 2 always pairs with B = 3. No counterexample appears, so the instance is CONSISTENT with A → B — but consistency in one instance is not proof for the schema.

3. Test A → C: the two tuples with A = 1 have C = 1 and C = 0 — a counterexample — so A → C is also refuted (though no option turns on it).

Why the other statements fail.

• Any statement asserting that B determines C contradicts the B = 1 counterexample directly.

• Any statement that includes 'A determines B' as a conclusion over-claims: that dependency is merely un-refuted here, not established for the schema.

• Any statement that A does NOT determine B is wrong too: no counterexample for A → B exists in the instance, so there is no basis to declare it false.

Result. The only conclusion the instance forces is that B does not functionally determine C.