Question $5$

$($a$)$ Use DeMorgan's laws to simplify the following expressions:

i$.\frac{?}{\text{b}\text{a}\text{r}}((\frac{?}{\text{b}\text{a}\text{r}}(a.b.(\frac{?}{\text{b}\text{a}\text{r}}(c))))\frac{+}{b}ar(((\frac{?}{\text{b}\text{a}\text{r}}(c))\frac{*}{b}ar())))$

ii$.\frac{?}{\text{b}\text{a}\text{r}}(()+b)\frac{*}{b}ar(b+(\frac{?}{\text{b}\text{a}\text{r}}(c)))\frac{*}{b}ar((\frac{?}{\text{b}\text{a}\text{r}}(c))+d)$

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$($b$)$ Given the followina loaic circuit:

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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

$($c$)$ Use the duality principle to find out the dual of the following equation:

$a*b+c\frac{*}{b}ar(d)=(a+c)*(a\frac{+}{b}ar(d))*(b+c)*(b\frac{+}{b}ar(d))$

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

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$F(a,b,c,d)=(+b\frac{*}{b}ar(d))*(c*b*a.\frac{+}{b}ar(c)*d)$

i$.$ The truth table

ii$.$ The Karnaugh map

iii. The minimal sum of products expression

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