7.5

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

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Proof using Armstrong’s axioms of the pseudotransitivity rule:

\[ \text{if $\alpha \rightarrow \beta$ and $\gamma\beta \rightarrow \delta$ then $\alpha\gamma \rightarrow \delta$ } \]

Proof:

\[ \alpha \rightarrow \beta \quad \text{given} \\ \alpha\gamma \rightarrow \gamma\beta \quad \text{augmentation rule and set union commutativity} \\ \gamma\beta \rightarrow \delta \quad \text{given} \\ \alpha\gamma \rightarrow \delta \quad \text{transitivity rule} \\ \]