Use DeMorgan$$s laws to simplify the following expressions:

i$.($a$.$b$.$c$)+($c$.$d$)$

ii$.$ a $+$ b $.$ b $+$ c $.$ c $+$ d

$($b$)$ Given the following logic circuit:

i$.$ Find a Boolean expression for the output Q for this logic circuit

ii$.$ Use the laws of boolean algebra and give to simplify this expression

Use the duality principle to find out the dual of the following equation:

a$.$b $+$ c$.$d $=($a $+$ c$).($a $+$ d$).($b $+$ c$).($b $+$ d$)$

$[2]$

$($d$)$ For the following Boolean expression, give:

F$($a$,$ b$,$ c$,$ d$)=($a $+$ b$.$d$).($c$.$b$.$a$.+$ c$.$d$)$

i$.$ The truth table

ii$.$ The Karnaugh map

iii. The minimal sum of products expression