B$1.$ You are require to design a logic controller. $A,B$ and C are three logic inputs and Y is the logic output of system. The requirement as following:

$(1)$ If A and C are both $0,Y=1.$

$(2)$ If A and B are both $1,Y=1.$

$(3)$ Others, $Y=0$

$($a$)$ Construct the truth table for Y$.$

$($b$)$ Develop the not simplified expression of Y in SOP form.

$($c$)$ Simplify Y by using K$-$map.

$(4$ marks$)$

$($d$)$ Convert ${11000}_{(2)}$ into a decimal number.

$(4$ marks$)$

$(8$ marks$)$

$(4$ marks$)$