GATE1988-12iia

Question

Using Armstrong’s axioms of functional dependency derive the following rules:

$\{ x \rightarrow y, \: x \rightarrow z \} \mid= x \rightarrow yz$

(Note: $x \rightarrow y$ denotes $y$ is functionally dependent on $x$, $z \subseteq y$ denotes $z$ is subset of $y$, and $\mid =$ means derives).

Answer 1

$x \rightarrow z$  (Given)

$\implies xx \rightarrow zx$ (Axiom of augmentation)  $\qquad \to(I)$

Also $x \rightarrow y$ (Given)

$\implies xz \rightarrow yz$ (Axiom of augmentation) $\qquad \to (II)$

Using $(I)$ and $(II)$ we get

$xx \rightarrow yz$ (Axiom of transitivity)

$\implies x \rightarrow yz$