$(25$ points$)$ Derive Decomposition from Armstrong's Axioms. I.e$.,$ assume AlongrightarrowBC, and prove both AlongrightarrowB and AlongrightarrowC by only using

Armstrong's Axioms. Each derivation step must indicate which rule is being used and on which antecedents.

$(25$ points$)$ Derive Pseudo$-$transitivity from Armstrong's Axioms. I.e$.,$ assume AlongrightarrowB and BClongrightarrowD, and prove AClongrightarrowD by only using

Armstrong's Axioms. Each derivation step must indicate which rule is being used and on which antecedents.

$(10$ points$)$ Let $A$ be a set of attributes, and let $F$ be a set of functional dependences. Module $2\mathrm{.}5$ Lecture $5$ presented a $O(|A||F|)-$time

algorithm to compute the closure of a set of attributes. The version here has been slightly optimized from the original by removing each functional

dependence from consideration once it has been applied. The changes are in bold. Briefly explain why this optimization doesn't change the

algorithm's correctness.

Closure $(A)=$

Result $=A$

Remaining $=F$

Repeat until Result doesn't change:

For each FD BlongrightarrowC in Remaining:

if Bsube Result,

then

$,$ Add $C$ to Result

Remove BlongrightarrowC from Remaining

Return Result