\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\vec}[1]{\mathbf{#1}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Section2.1Mathematical expressions using LaTeX

Mathematical content should be displayed using LaTeX. This is accomplished using either <m></m> for inline expressions or <me></me> for display expressions.

For those who are new to LaTeX, several types of expressions appear in this section. The source code for this document can be consulted to see the LaTeX code to realize them.

Note2.1.1

  1. The result of doing arithmetic modulo \(n\) is always an integer between 0 and \(n-1\), by the Division Property. This observation implies that \(\{0, 1,\dots, n-1\}\) is closed under modulo \(n\) arithmetic.

  2. It is always true that \(a +_n b \equiv (a + b) (\textrm{ mod} n)\) and \(a\times _n b \equiv (a \cdot b) (\textrm{ mod} n)\). For example, \(4 +_7 5 = 2 \equiv 9 (\mod 7)\) and \(4 \times _7 5 \equiv 6 \equiv 20 (\mod 7)\).

  3. We will use the notation \(\mathbb{Z}_n\) to denote the set \(\{0, 1, 2,. . ., n-1\}\).

A simple array of sets.

\begin{equation*} \begin{array}{cc} \{1,2\} & \{3,4\} \\ \{1,3\} & \{2,4\} \\ \{1,4\} & \{2,3\} \\ \{2,3\} & \{1,4\} \\ \{2,4\} & \{1,3\} \\ \{3,4\} & \{1,2\} \\ \end{array} \end{equation*}\begin{equation*}\binom{5}{0} x^5+\binom{5}{1} x^4 y+\binom{5}{2} x^3 y^2+\binom{5}{3} x^2 y^3+\binom{5}{4} x \binom{5}{5} y^4+y^5 \end{equation*}

Here is a chain of equalities:

\begin{equation*}\begin{split} \lvert A \rvert & = \lvert A_1 \cup A_2 \cup A_3 \rvert \\ & = \lvert A_1 \rvert + \lvert A_2 \rvert + \lvert A_3 \rvert - \textrm{repeats} \\ & = \lvert A_1 \rvert + \lvert A_2 \rvert + \lvert A_3 \rvert - \textrm{duplicates} + \textrm{triplicates} \\ & = \lvert A_1 \rvert + \lvert A_2 \rvert + \lvert A_3 \rvert - (\lvert A_1 \cap A_2 \rvert + \lvert A_1 \cap A_3 \rvert+ \lvert A_2 \cap A_3 \rvert) + \lvert A_1 \cap A_2 \cap A_3 \rvert \\ & = 75 + 60 + 55 - 25 - 12 - 15 + 10 = 148 \\ \end{split} \end{equation*}

Using md: \begin{equation*} \begin{split} T(n) &= T\left(a_1a_2\ldots a_r\right)\\ & =1+ T\left(a_1a_2\ldots a_{r-1}\right)\quad \\ & =1+\left(1+ T\left(a_1a_2\ldots a_{r-2}\right)\right)\\ & =2+ T\left(a_1a_2\ldots a_{r-2}\right)\\ &\quad \vdots \\ & = (r-1) + T\left(a_1\right)\\ & = (r-1)+1\quad \textrm{ since } T(1)=1\\ & = r\\ \end{split} \end{equation*}

Here's a truth table generated in LaTeX with array:

\begin{equation*} \begin{array}{cccc} p & q & \neg (p\land q) & \neg p\lor \neg q \\ \hline 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ \end{array} \end{equation*}\begin{equation*}\begin{array}{cc} p & \neg p \\ \hline 0 & 1 \\ 1 & 0 \\ \end{array} \end{equation*}

... and with table

\begin{equation*} \begin{table} \begin{tabular}{ccc} p & q & p\land q \\ 0 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \\ \end{tabular} \caption{Truth table for conjunction} \end{table} \end{equation*}

The table for \(\oplus\) is \[ \begin{array}{c | c c c} \oplus & 0 & 1 & 2 \\ \hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \end{array} \]

Commutative Laws \begin{gather} \(a\lor b = b\lor a\)\label{boolean-law-1}\tag{2.1.1}\\ \(a \land b = b \land a\) \label{boolean-law-1-prime}\tag{2.1.2} \end{gather}
Associative Laws \begin{gather} \(a \lor (b \lor c) = (a \lor b) \lor c\) \label{boolean-law-2}\tag{2.1.3}\\ \(a \land (b \land c) = (a \land b) \land c\)\label{boolean-law-2-prime}\tag{2.1.4} \end{gather}
Distributive Laws \begin{gather} \(a \land (b \lor c) = (a \land b) \lor (a \land c)\text \) \label{boolean-law-3}\tag{2.1.5}\\ \(a \lor (b \land c) = (a \lor b) \land (a \lor c)\)\label{boolean-law-3-prime}\tag{2.1.6} \end{gather}
Identity Laws \begin{gather} \(a \lor 0 = 0 \lor a = a\) \label{boolean-law-4}\tag{2.1.7}\\ \(a \land 1= 1 \land a = a\)\label{boolean-law-4-prime}\tag{2.1.8} \end{gather}
Complement Laws \begin{gather} \(a \lor \bar{a} = 1\) \label{boolean-law-5}\tag{2.1.9}\\ \(a \land \bar{a}= 0\)\label{boolean-law-5-prime}\tag{2.1.10} \end{gather}
Idempotent Laws \begin{gather} \(a \lor a = a\) \label{boolean-law-6}\tag{2.1.11}\\ \(a \land a = a\)\label{boolean-law-6-prime}\tag{2.1.12} \end{gather}
Null Laws \begin{gather} \(a \lor 1 = 1\) \label{boolean-law-7}\tag{2.1.13}\\ \(a \land 0 = 0\) \label{boolean-law-7-prime}\tag{2.1.14} \end{gather}
Absorption Laws \begin{gather} \(a \lor (a \land b) = a\) \label{boolean-law-8}\tag{2.1.15}\\ \(a \land (a \lor b) = a\) \label{boolean-law-8-prime}\tag{2.1.16} \end{gather}
DeMorgan's Laws \begin{gather} \(\overline{a \lor b} = \bar{a} \land \bar{b}\) \label{boolean-law-9}\tag{2.1.17}\\ \(\overline{a \land b} = \bar{a} \lor \bar{b}\) \label{boolean-law-9-prime}\tag{2.1.18} \end{gather}
Involution Law \begin{gather} \(\overline{\bar{a}} = a\)\label{boolean-law-10}\tag{2.1.19} \end{gather}
Table2.1.3Basic Boolean Algebra Laws