$($ii$)$ Let $x=\{a,b,c,d\}$ and $Y=\{1,2,3,4\}.$

tilde$(A)=\frac{0\mathrm{.}1}{a}+\frac{0\mathrm{.}8}{b}+\frac{0\mathrm{.}6}{c}+\frac{1}{d},$tilde$(B)=\frac{0\mathrm{.}2}{a}+\frac{1}{b}+\frac{0\mathrm{.}9}{c}+\frac{0\mathrm{.}1}{d}$ and tilde$(C)=\frac{0\mathrm{.}3}{1}+\frac{0\mathrm{.}4}{2}+\frac{1}{3}+\frac{0\mathrm{.}7}{4}$

Determine the implication relations

$($a$)$ If $x$ is tilde$(A)$ then $z$ is tilde$(C)$

$($b$)$ If $x$ is tilde$(A)$ and $y$ is tilde$(B)$ then $z$ is tilde$(C)$