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. A single relation instance can be used to DISPROVE a dependency — one pair of tuples that agree on X but differ on Y is a decisive counterexample — but it can never PROVE a dependency for the schema, because some unseen legal tuple might still violate it. So from one instance, the only certain conclusions are the dependencies it refutes.

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.