7.27

Use Armstrong’s axioms to prove the soundness of the decomposition rule.

---

• Decomposition Rule states: If \(\alpha \rightarrow \beta\gamma\) holds, then \(\alpha \rightarrow \beta\) holds and \(\alpha \rightarrow \gamma\) holds.

Suppose \(\alpha \rightarrow \beta\gamma\) holds. By Reflexivity rule we know that, \(\beta\gamma \rightarrow \beta\) and \(\beta\gamma \rightarrow \gamma\) holds.

By Transitivity rule \(\alpha \rightarrow \beta\) holds and \(\alpha \rightarrow \gamma\) holds.