A flow rate sensing device used on a liquid transport pipeline functions as follows. The device provides a $5-$bit output where all five bits are zero if the flow rate is less than $10$ gallons per minute. The first bit is $1$ if the flow rate is at least $10$ gallons per minute; the first and second bits are $1$ if the flow rate is at least $20$ gallons per minute; the first, second, and third bits are $1$ if the flow rate is at least $30$ gallons per minute; and so on$.$ The five bits, represented first through fifth by the logical variables A$,$ B$,$ C$,$ D$,$ and E$,$ respectively, are used as inputs to a device that provides two outputs Y and Z$.($Use the following as necessary: A$,$ B$,$ C$,$ D$,$ and E$.)$

$($a$)$

Write an equation for the output Y if we want Y to be $1$ if the flow rate is less than $40$ gallons per minute.

Y $=$

Correct: Your answer is correct.

$($b$)$

Write an equation for the output Z if we want Z to be $1$ if the flow rate is at least $10$ gallons per minute but less than $40$ gallons per minute.

Z $=$

Incorrect: Your answer is incorrect.