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%\usepackage{opensans}          % Can't make this work so far. Shame. Would be lovely.
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% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
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% Document Info
\author{ozzyfisk0}
\pdfinfo{
  /Title (logic-and-computing-devices.pdf)
  /Creator (Cheatography)
  /Author (ozzyfisk0)
  /Subject (Logic and Computing Devices Cheat Sheet)
}

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% Header and Footer
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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
    \SetRowColor{DarkBackground}
    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Logic and Computing Devices Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{ozzyfisk0} via \textcolor{DarkBackground}{\uline{cheatography.com/105592/cs/21392/}}}
\end{tabulary}
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\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}ozzyfisk0 \\
  \uline{cheatography.com/ozzyfisk0} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 12th December, 2019.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
\raggedright
\raggedcolumns

% Set font size to small. Switch to any value
% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{p{2.508 cm} p{3.04 cm} p{2.052 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Common Conversions}}  \tn
% Row 0
\SetRowColor{LightBackground}
2\textasciicircum{}10\textasciicircum{} & 10\textasciicircum{}3\textasciicircum{} & Kilo \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
2\textasciicircum{}20\textasciicircum{} & 10\textasciicircum{}6\textasciicircum{} & Mega \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
2\textasciicircum{}30\textasciicircum{} & 10\textasciicircum{}9\textasciicircum{} & Giga \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
2\textasciicircum{}40\textasciicircum{} & 10\textasciicircum{}12\textasciicircum{} & Tera \tn 
% Row Count 4 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{5.44 cm} x{2.56 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Information Display}}  \tn
% Row 0
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K & 10\textasciicircum{}3\textasciicircum{} bytes \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Kib, KB (i=information) & 2\textasciicircum{}10\textasciicircum{} bytes \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Number Formulas}}  \tn
% Row 0
\SetRowColor{LightBackground}
Max value represented by an {\bf{n}} bit number & 2\textasciicircum{}{\bf{n}}\textasciicircum{}-1 \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Max value in range of fractions & (2\textasciicircum{}{\bf{n}}\textasciicircum{}-1)/2\textasciicircum{}{\bf{n}}\textasciicircum{} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
{\bf{n}} number of bits to represent number {\bf{x}} & log({\bf{x}})/log(2) \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
Digits to store {\bf{n}} in binary & log({\bf{n}})/log(2) \tn 
% Row Count 10 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Digits to store {\bf{b}} bit binary number in BCD & 4x{[}log(2\textasciicircum{}{\bf{b}}\textasciicircum{}){]} \textless{}-brackets are rounded up \tn 
% Row Count 13 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Complements}}  \tn
% Row 0
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(r-1)'s Complement of {\bf{n}}, where {\bf{d}} = number of digits & (r\textasciicircum{}{\bf{d}}\textasciicircum{}-1)-{\bf{n}} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
1's Complement & Complement each bit \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
2's Complement & Start from right until 1. Then, take complement. \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
For base R, R's complement of n & r\textasciicircum{}d\textasciicircum{}-n \tn 
% Row Count 9 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Signed Arithmetic}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{If the value is negative, take the 2's complement. Then, add both values together.% Row Count 2 (+ 2)
} \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.32 cm} x{5.68 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Sampling temps from -x to y. {\bf{n}} bits per sample}}  \tn
% Row 0
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y-{}-x = {\bf{Z}} & log({\bf{Z}})/log({\bf{n}}) = {\bf{F}} \tn 
% Row Count 2 (+ 2)
% Row 1
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\mymulticolumn{2}{x{8.4cm}}{Then use {\bf{F}} as a base to find "binary" value} \tn 
% Row Count 3 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.6 cm} x{4.4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Boolean Algebra Rules}}  \tn
% Row 0
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X+1=1 & Annulment \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
X+0=0 & Identity \tn 
% Row Count 2 (+ 1)
% Row 2
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X*1=X & Identity \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
X*0=0 & Annulment \tn 
% Row Count 4 (+ 1)
% Row 4
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X+X=X & Idempotent \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
X*X=X & Idempotent \tn 
% Row Count 6 (+ 1)
% Row 6
\SetRowColor{LightBackground}
(X')' = X & Double Negation \tn 
% Row Count 7 (+ 1)
% Row 7
\SetRowColor{white}
X+X' = 1 & Complement \tn 
% Row Count 8 (+ 1)
% Row 8
\SetRowColor{LightBackground}
X*X'=0 & Complement \tn 
% Row Count 9 (+ 1)
% Row 9
\SetRowColor{white}
X+Y=Y+X & Commutative \tn 
% Row Count 10 (+ 1)
% Row 10
\SetRowColor{LightBackground}
X{\emph{Y=Y}}X & Commutative \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
X'+X' = X'X' & deMorgan's Theorem \tn 
% Row Count 12 (+ 1)
% Row 12
\SetRowColor{LightBackground}
X'X' = X'+X' & deMorgan's Theorem \tn 
% Row Count 13 (+ 1)
% Row 13
\SetRowColor{white}
X+XY = X & Absorption \tn 
% Row Count 14 (+ 1)
% Row 14
\SetRowColor{LightBackground}
X(X+Y) = X & Absorption \tn 
% Row Count 15 (+ 1)
% Row 15
\SetRowColor{white}
Associative Law & Distributive Law \tn 
% Row Count 16 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{Can think of AND as series, and OR as parallel}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Cost Criteria}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{literals+non-single terms + unique complemented literals} \tn 
% Row Count 2 (+ 2)
% Row 1
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\mymulticolumn{1}{x{8.4cm}}{Ex: ABCD+A'B'C'D'} \tn 
% Row Count 3 (+ 1)
% Row 2
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\mymulticolumn{1}{x{8.4cm}}{(8) + (2) + (4) = 14} \tn 
% Row Count 4 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Flip Flop Characteristic Tables}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/ozzyfisk0_1576190936_Screenshot_2019-12-12 ECE2010Slides(Chapter5 4-FlipFlops) pptx S1908-ECE-2010 Logic and Computing Devices - 001.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Sequential Analysis}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Write all of the inputs for each flip-flop. Make a state table.% Row Count 2 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{State Reduction}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{If two states have the same inputs and outputs, you can remove one. Remember to change the variable if it appears elsewhere.% Row Count 3 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.76 cm} x{4.24 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Counters}}  \tn
% Row 0
\SetRowColor{LightBackground}
Synchronous Counters & have a common clock \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Ring counter & circular shift register \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4.56 cm} x{3.44 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Types of PLD's}}  \tn
% Row 0
\SetRowColor{LightBackground}
No fishbones, 4x16 decoder & 16x8 ROM \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Fishbones, x's & PLA (Programmable Logic Array) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
No fishbones, 2x4 decoder & 4x2 ROM \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Fixed ORs, same inputs & PAL (Programmable Array Logic) \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
When finding terms, if X on 1, NOT all terms & Else, find terms normally \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.04 cm} x{4.96 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Parity}}  \tn
% Row 0
\SetRowColor{LightBackground}
EVEN Function & generates ODD parity \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
ODD Function & generates EVEN parity \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}