Important: Homework is an individual work and each student is expected do his own work. Note letters such as$,$ B$,$ a$,$ b$,$ c$,$ d$,$ e$,$ etc., represent Boolean variables, unless stated otherwise.

$($Make sure you show all steps.$)$

Total $=100$ Points

PART A

$(40$ points$)$

Using Boolean algebra, prove that manipulation:

$\frac{?}{\text{b}\text{a}\text{r}}(A)B\frac{+}{b}ar\frac{B}{b}ar(C)+AB\frac{+}{b}ar(B)C=1.$

$(5pts)$

$2.$ We have a computer that stores binary signed integers in two's complement form. All numbers are $8$ bits long.

$(10$ pts$)$

a$.$ What decimal number is represented by $01101011?$

b$.$ What decimal number is represented by $10101110?$

c$.$ How is the number $-113$ stored?

d$.$ How is the number $+143$ stored?