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I am trying to figure out how I can turn this question into a Boolean function (which will then be turned into a boolean circuit). I know that $-6$ in $4$-bit two's complement is 1010 and I know the two's complements that are less than $-6,$ but I have no idea how I would go about writing this as a function as I've only worked with boolean formulas such as: ¬A→(¬B∧¬C) etc

DanielV
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mdbej
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  • 4 bits is small enough that a good place to start might be to just write out which things should evaluate to true, and which things to false. – Misha Lavrov Aug 27 '20 at 16:44

1 Answers1

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$-6$ is $1010$, $-7$ is $1001$, $-8$ is $1000$; all others are bad for our purposes, so we have, with the bits labeled $abcd$,

$$\begin{align} &ab'cd' + ab'c'd + ab'c'd'\\ =\,&ab'cd' + ab'c'd + ab'c'd' + ab'c'd'\\ =\,&ab'cd' + ab'c'd' + ab'c'd + ab'c'd'\\ =\,&ab'd'(c+c') + ab'c'(d+d')\\ =\,&ab'd'+ab'c'\\ =\,&ab'(c'+d')\\ =\,&ab'(cd)' \end{align}$$

and any of those last three I'd call simplified.

Dan Uznanski
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