$12\mathrm{.}0.$ Consider the following four$-$bit numbers, with decimal values in parentheses:

$1000(8)$

$1011(11)$

$1101(13)$

$1110(14)$

The ones$-$complement sum of these can be found using the division method by treating these as a four$-$digit hex number $0$x$8$bde and taking the remainder mod $15$; the result is $1.$

$($a$).$ Find this ones$-$complement sum via three $4-$bit ones$-$complement additions. To get started, note that the $($exact$)$ sum of $1000$ and $1011$ is $1|0011,$ and adding the carry bit to the low$-$order $4$ bits gives a ones$-$complement sum of the first pair of $0100.$

$($b$).$ The exact $($and $8-$bit twos$-$complement$)$ sum of the values above is $46,$ or $10|1110$ in binary. Find the ones$-$complement sum of the values by taking this exact sum and then forming the ones$-$complement sum of the $4-$bit high and low halves. Note that this is not the same as the twos$-$complement sum of the halves.