$3.[$Functional Dependencies$](30)$ Recall Armstrong$$s axioms.

$$ Reflexivity rule: if Y $$ X then X $->$ Y

$$ Augmentation rule: if X $->$ Y then XZ $->$ YZ

$$ Transitivity rule: if X $->$ Y and Y $->$ Z then X $->$ Z

a$.[10$ pts$]$ Give two examples for using Armstrong$$s inference rules to induce new FDs from the set of FDs you designed in question $2($b$).$

b$.[20$ pts$]$ Prove the following inference rules also hold, using FD definition and Armstrong$$s Axioms.

i$.$ decomposition rule: if X $->$ YZ then: X $->$ Y and X $->$ Z

ii$.$ Psuedo transitivity: if X $->$ Y and YW $->$ Z then: XW $->$ Z