Consider a relation R(A, B, C). Which of the following statement is not true according to inference rules for functional dependencies ?

Question

Accepted Answer

The correct option is (2):
The statement "If AB → C, then A → B and B → C" is **not true** according to the inference rules for functional dependencies (FDs), also known as Armstrong's Axioms. The fundamental principle of FDs is that you can decompose the *right-hand side* (the dependent attributes), but **not** the *left-hand side* (the determinant attributes). The functional dependency $AB \to C$ only guarantees that the composite attribute $AB$ uniquely determines $C$. It makes absolutely no assertion about whether $A$ independently determines $B$ (i.e., $A \to B$) or whether $B$ independently determines $C$ (i.e., $B \to C$). For instance, in a table with attributes (FirstName, LastName, Course), the combination (FirstName, LastName) might determine the Course, but neither FirstName alone determines LastName, nor does LastName alone determine Course.

The incorrect option is (1):
The statement "If A → B and B → C, then A → C" represents the **Transitivity Rule**, which is one of the primary axioms in Armstrong's set of inference rules. This rule is fundamentally sound and is always true. If attribute A uniquely determines attribute B, and B in turn uniquely determines attribute C, it logically follows that A must also uniquely determine C. This is a vital property for determining closure and identifying keys in a relational schema.

The incorrect option is (3):
The statement "If A → B and A → C, then A → BC" is an application of the **Union Rule**. This is a correct and valid inference rule derived from Armstrong's axioms. It states that if a single attribute or set of attributes ($A$) determines multiple other attributes ($B$ and $C$) separately, then that same attribute set ($A$) must also determine the combination (or union) of those dependent attributes ($BC$). This rule is essential for simplifying sets of functional dependencies.

The incorrect option is (4):
The statement "If A → B, then AC → BC" is an application of the **Augmentation Rule**, which is also an axiom of Armstrong's system and is always true. This rule permits the addition of the exact same attribute or set of attributes ($C$) to the right-hand side and the left-hand side of an existing functional dependency ($A \to B$). Since $A$ alone determines $B$, adding $C$ to $A$ (creating $AC$) will still determine $B$, and since $C$ is always determined by itself, $AC$ determines $BC$.