Question $4$

$[15$ marks$]$

$4($a$)$ Simplify $\backslash ($ F$($A$,$ B$,$ C$)=\backslash$bar$\{$A$\}$ B C$+$A $\backslash$bar$\{$B$\}$ C$+$A B $\backslash$bar$\{$C$\}+$A B C $\backslash )$ by Karnaugh map. $(5$ marks$)$

$4($b$)$ As an example of a combinational circuit, we learnt the logic of a half adder in Lecture $12.$ By connecting two half$-$adders, a full adder circuit is constructed. A complete adder includes three binary digit inputs $($two input bits and one carry input bit$),$ as well as two binary digit outputs: the sum bit $($ S $)$ and the carry output bit $($Cout$)$:

$($i$)$ Determine the truth table of a full adder logic. $(4$ marks$)$

$($ii$)$ Determine the logic of Sum output bit of a full adder. $(2$ marks$)$

$($iii$)$ Determine the logic of Carry output bit of a full adder. $(2$ marks$)$

$($iv$)$ Draw the combinational circuit of the full adder. $(2$ marks$)$