NAND and NOR are **universal gates** — some combination of them can form any other logic gate. Constructions of other gates using only these gates are called **NAND-NAND realisations** or **NOR-NOR realisations**.
This is useful in SOP as if two ANDs feed into an OR, all can be turned into NANDs to achieve the same result.
!!! example
NOT can be expressed purely with NAND as $A$ NAND $A$:
The **synthesis** of an algebraic formula represents its implementation via logic gates. In this course, its total cost is the sum of all inputs to all gates and the number of gates, *excluding* initial inputs of "true" or an initial negation.
In order to deduce an algebraic expression from a truth table, **OR** all of the rows in which the function returns true and simplify.
The **minterm** $m$ is a **product** term where all variables in the function appear once. There are $2^n$ minterms for each function, where $n$ is the number of input variables.
To determine the relevant function, the subscript can be converted to binary and each function variable set such that:
- if the digit is $1$, the complement is used, and
- if the digit is $0$, the original is used.
$$m_j=x_1+x_2+\dots x_n$$
!!! example
For a function that accepts three variables:
- there are eight minterms, from $m_0$ to $m_7$.
- the sixth minterm $m_6=xyz'$ because $6=0b110$.
For a sample function defined by the following minterms:
$$
\begin{align*}
f(x_1,x_2,x_3)&=\sum m(1,2,5) \\
&=m_1+m_2+m_5 \\
&=x_1x_2x_3' + x_1x_2'x_3 + x_1'x_2x_3'
\end{align*}
$$
The **maxterm** $M$ is a **sum** term where all variables in the function appear once. It is more or less the same as a minterm, except the condition for each variable is **reversed** (i.e., $0$ indicates the complement).
$$M_j=x_1+x_2+\dots +x_n$$
!!! example
For a sample function defined by the following maxterms:
\begin{align*}
f(x_1,x_2,x_3,x_4)&=\prod M(1,2,8,12) \\
&=M_1M_2M_8M_{12} \\
\end{align*}
??? example
Prove that $\sum m(1,2,3,4,5,6,7)=x_1+x_2+x_3$: **(some shortcuts taken for visual clarity)**
Binary is represented in hardware via switches called **transistors**. Above a certain voltage threshold, its output is $1$, whlie it is $0$ if below a threshold instead.
A transistor has three inputs/outputs:
- A ground
- An input **source**, which has voltage that determines whether the circuit is connected to the ground
- An output **drain**, which will either be grounded or have a voltage depending on whether the switch is closed.
