For a relation scheme R(ABC), assume that the attributes are prime attributes,…

Concept

A prime attribute is one that belongs to at least one candidate key; a non-prime attribute belongs to none. The normal forms differ in what they demand of every non-trivial functional dependency X → A. 3NF requires that, for each such FD, X is a superkey OR A is a prime attribute. BCNF drops the second escape clause: it requires X to be a superkey for every non-trivial FD, with no exception for prime A.

Application

1. Suppose every attribute of R(A, B, C) is prime, so the relation has no non-prime attribute at all.

2. Take any non-trivial FD X → A that holds on R. Its right-hand side A is an attribute of R, and by assumption A is prime.

3. The 3NF rule offers two ways to be satisfied: X is a superkey, or A is prime. Here the second clause holds for every FD because A is always prime, so the 3NF condition is met automatically.

4. 2NF and 1NF are weaker requirements implied by 3NF, so they also hold. 1NF needs only atomic values; 2NF forbids a non-prime attribute depending on part of a key, which cannot arise when there are no non-prime attributes.

Cross-check

BCNF is not guaranteed: it ignores whether A is prime and insists X be a superkey. A relation can have an FD whose left side is not a superkey while its right side is prime. Concrete witness: R(A, B, C) with candidate keys AB and AC and the FD C → B. Every attribute is prime, yet C is not a superkey, so this FD breaks BCNF while still passing 3NF. Hence the strongest form guaranteed purely from 'all attributes prime' is 3NF, not BCNF.