$(15$ points$)$ A sender needs to send out $4$ numbers $(\backslash (3,5,11,9\backslash )).$ A checksum number will be sent together with the $4$ numbers for error detection. Answer the following questions:

a$.$ Find the checksum calculated at the sender side

b$.$ If there is no error during the transmission, what is the checksum calculated at the receiver side?

c$.$ If the first number $3$ is changed to $1(0011$ is changed to $0001)$ because of transmission error, what is the checksum calculated at the receiver side $($receiver receives $\backslash (1,5,11,9\backslash ))?$

d$.$ If the first data is changed from $3$ to $1,$ and the second data is changed from $5$ to $7,$ what is the checksum calculated at the receiver side $($receiver receives $\backslash (1,7,11,9\backslash ))?$