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:

$$\text{if α→β and γβ→δ then αγ→δ }$$

Proof:

$$\begin{gathered} \alpha \to \beta \text{given} \\ \alpha \gamma \to \gamma \beta \text{augmentation rule and set union commutativity} \\ \gamma \beta \to \delta \text{given} \\ \alpha \gamma \to \delta \text{transitivity rule} \end{gathered}$$