Challenge Sheet $2$

Logic Circuits and Boolean Algebra

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$1.$ Design a Comparator circuit in logisim that compares the binary values of two $2-$bit integers A$($a$1,$a$0)$ and B$($b$1,$b$0)$ and generates a signal on one of its outputs, indicating whether A$>$B$,$ AB $=1,$ or if A$=(1,0)$ and B$=(1,1)$ then output wire AB and AB and A$=$B$.$

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f$.$ Using the equations:

XY $+$ XY$=$ X$($Y$+$Y$)=$ X$\mathrm{.}1=$ X

XYZ $+$ XYZ$+$ XY$$Z $+$ XY$$Z$=$ X$($YZ $+$ YZ$+$ Y$$Z $+$ Y$$Z$)=$ X$\mathrm{.}1=$ X

$$ Can the equations for AB be simplified $?$ If so$,$ what are these$$

$$ equations and what are

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$2.$ The following circuit :

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has the Boolean equation:

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$($A $+$ B$+$ C$)+[($C $+$ B$).($AB$)]+0$

$=($A $+$ B$+$ C$)+[($C $+$ B$).($AB$)]$

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a$.$ Use De$$ Morgan$$s Theorem to expand and simplify it as far as possible. $($You can repeatedly apply De Morgan$$s law.

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b$.$ What is the equivalent gate circuit of the equation that you have produced in $2.$a $?$

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