$1.$ If a relation R is decomposed into $\{$R$1,$ R$2,...,$ Rn$\}$ and the decomposition is lossless then

$($a$)$ The natural join of R$1,$ R$2,...,$ Rn can have more tuples than the original relation for R

$($b$)$ The relations R$1,$ R$2,...,$Rn are each in $3$rd Normal Form

$($c$)$ The natural join of R$1,$ R$2,...,$ Rn will have the same number of tuples as the original relation R

$($d$)$ None of the above

$2.$The functional dependency $\{$A$\}\{$B$\}$ for relation schema R$($A$,$B$,$C$,$D$)$ implies that

$$

$($a$)$ No two tuples in R can have the same value for attribute B

$($b$)$ Any two tuples in R that have the same value for B must have the same value for A

$($c$)$ No two tuples in R can have the same value for attribute A

$($d$)$ Any two tuples in R that have the same value for A must have the same value for B