7.4

Use Armstrong’s axioms to prove the soundness of the union rule. (Hint: Use the augmentation rule to show that, if $\alpha \to \beta$, then $\alpha \to \alpha \beta$. Apply the augmentation rule again, using $\alpha \to \gamma$, and then apply the transitivity rule.)

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To prove that:

$$\text{if α→β and α→γ then α→βγ }$$

Following the hint, we derive:

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