A paper tape reader used as an input device to a computer has five rows of holes as shown. A hole punched in the tape indicates a logic $1,$ and no hole indicates a logic $0.$ As each hole pattern passes under the photocells, the pattern is translated into logic signals on lines $A,B,C,D,$ and $E.$ All patterns of holes indicate a valid character with two exceptions. A pattern consisting of none of the possible holes punched is not

used because it is impossible to distinguish between this pattern and the unpunched space between patterns. An incorrect pattern punched on the tape is erased by punching all five holes in that position. Therefore, a valid character punched on the tape will have at least one hole but will not have all five holes punched.

$($a$)$ Write an equation for a variable $Z$ which is $1$ iff a valid character is being read.

$($b$)$ Write an equation for a variable $Y$ which is $1$ iff the hole pattern being read has holes punched only in rows $C$ and $E.$