Homework # $8$

$1.$ Consider the following relation r below.

r

A B C D E

a$1$ b$1$ c$1$ d$1$ e$1$

a$1$ b$2$ c$2$ d$2$ d$1$

a$2$ b$1$ c$3$ d$3$ e$1$

a$2$ b$1$ c$4$ d$3$ e$1$

a$3$ b$2$ c$5$ d$1$ e$1$

Which of the following FDs does r satisfy?

A $$ D$,$AB $$ D$,$C $$ BDE,E $$ A$,$A $$ E

$2.$ Let F $=\{$AB$$C$,$ B$$D$,$ CD$$E$,$ CE$$GH$,$ G$$A$\}.$

Using Armstrong$$s axioms, give proofs f or the following:

a$.$ F$$AB$$E

b$.$ F $$ BG $$ C

c$.$ F$$AB$$G

$3.$ Prove or disprove the following inference rules for a relation r$($R$)$ with W$,$ X$,$ Y$,$ and Z subsets of attributes of R$.$

a$.$ X $$ Y and Z $$ W imply XZ $$ YW$.$

b$.$ XY $$ Z and Z $$ X imply Z $$ Y$.$

c$.$ X $$ Y and Y $$ Z imply X $$ Y Z$.$