A computer interface to a line printer has seven data lines that control the movement of the paper and the print head and determine which character to print. The data lines are labeled $A,B,C,D,E,F,$ and $G,$ and each represents a binary $0$ or $1.$ When the data lines are interpreted as a $7-$bit binary number with line A being the most significant bit, the data lines can represent the numbers $0$ to ${127}_{10}.$ The number ${13}_{1n}$ is the command to return the print head to the beginning of a line, the number ${10}_{10}$ means to advance the paper by one line, and the numbers ${32}_{1n}$ to ${127}_{10}$ represent printing characters.

$($a$)$ Write an equation for the variable $x$ which is $1$ iff the data lines indicate a command to return the print head to the beginning of the line.

$($b$)$ Write an equation for the variable $Y$ which is $1$ iff there is an advance paper command on the data lines.

$($c$)$ Write an equation for the variable $7$ which is $1$ iff the data lines indicate a printable character. $($Hint: Consider the binary representations of the numbers $0-31$ and $32-127$ and write the equation for $7.$ with only two terms.$)$