$1.(20$ pts$)$ Consider keyed functions Fk that is to perform OR operation on a two$-$bit input x$\{0,1\}2$ with a key k$\{0,1\}2$ to generate an output y$,$ i$.$e$.,$ y$=$Fk$($x$)=$k OR x$.$ Please address the following:

$($a$)$ How many $($and why so$)$ functions are in the set of random functions f and the set of keyed functions Fk respectively? Please show each Fk by the complete mapping from X to Y$.$ Please identify one function $($the complete mapping from X to Y$)$ that is unique to be in one set but not the other set.

$($b$)$ Please give out the design of an efficient distinguisher D and calculate the following difference for this distinguisher D as in the definition of PRF$.$ You must show how to calculate it clearly as we did in class.

$|$ Pr$[(.)(12)=1]$ Pr$[(.)(12)=1]|$