f$)$ Consider the following problems that need to be solved using a Genetic Algorithm:

$($i$)$ Find a $20-$bit binary string that contains exactly five $1$s$.$

$($ii$)$ Find the integer coefficients for xi

that will solve the following equation:

x$1+2$x$2+3$x$3+4$x$4=30$

$($iii$)$ Find all the real$-$valued intersection points of the following three functions:

f$1($x$)=$ x $+1,$ f$2($x$)=$ x

$2$

$,$ f$3($x$)=$ x

$5+1$

$($iv$)$ Find real$-$valued solutions for x and y in the equation:

$2$x

$24$x $+$ y $=1$

For each of these tasks:

$$ Design an appropriate chromosome to represent solutions to the task. Write down your

definition of the chromosome, using appropriate terminology.

$$ Design a fitness function, keeping the criteria for good fitness functions in mind. Write

down your definition of the fitness function, using appropriate terminology.