\documentclass[
	nosolutions,
	singlenumbering,
	nopagenumber
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}


\title{Warm-Up: Adders}
\subtitle{Prepared by \githref{Mark} on \today}

\begin{document}

	\maketitle

	\problem{}
	Fill the following binary addition table. \par
	\hint{s is \say{sum,} c is \say{carry}}

	\begin{center}
		\begin{tabular}{ c c || c c }
			$a$ & $b$ & s & c \\
			\hline
			0 & 0 & ? & ? \\
			0 & 1 & ? & ? \\
			1 & 0 & ? & ? \\
			1 & 1 & ? & ?
		\end{tabular}
	\end{center}

	\vfill

	\problem{}
	Draw a logic circuit that atisfies the above table. \par
	This is called a \textit{half adder}. \par
	\hint{You should need exactly two gates.}

	\begin{solution}
		$s = a \texttt{ xor } b$ \par
		$c = a \texttt{ and } b$
	\end{solution}

	\vfill

	\definition{}
	A \textit{full adder} is similar to a half adder, but it has an extra input: \par
	a full adder takes $a$, $b$, and $c_\text{in}$, and produces $s$ and $c_\text{out}$. \par
	\hint{$c_\text{in}$ is \say{carry in}}

	\problem{}
	Use two half adders to construct a full adder.

	\begin{solution}
		$s_1, c_1 = \texttt{HA}(a, b)$ \par
		$s_2, c_2 = \texttt{HA}(s_1, c_\text{in})$ \par
		$s_\text{out} = s_2$ \par
		$c_\text{out} = \texttt{OR}(c_1, c_2)$

		\vspace{2mm}

		Of course, the class should just draw the circuit.
	\end{solution}


	\vfill
	\pagebreak

	\problem{}<rippleadder>
	How can we add two four-bit binary numbers using the full adder? \par
	We want a four-bit output sum and a one-bit $c_\text{out}$.

	\vfill

	\problem{}
	Say that all basic logic gates need $1u$ of time to fully switch states. \par
	\note[Note]{This is called \textit{gate delay}}

	\vspace{2mm}

	How much time does a full adder need to fully switch states? \par
	How about your circuit from \ref{rippleadder}?

	\vfill

	\problem{Bonus}
	Design a faster solution to \ref{rippleadder}.


	\vfill
	\pagebreak

\end{document}