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\begin(align*)
\overline{A \cap B \cap C} &\equiv x \notin A \cap B \cap C & \text{definition of complement} \\
&\equiv \neg x \in A \cap B \cap C &\text{definition of does not belong symbol} \\
&\equiv \neg(x \in A \land x \in B \land x\in C) & \text{definition of intersection} \\
&\equiv \neg(x \in A) \lor \neg (x \in B) \lor \neg (x \in C) & \text{by 1st DeMorgan's Law for logical equiv.} \\
&\equiv x \notin A \lor x \notin B \lor x \notin C & \text{by definition of does not belong symbol} \\
&\equiv x \in \overline{A} \lor x \in \overline{B} \lor x \in \overline{C} &\text{definition of complement} \\
&\equiv x \in (\overline{A} \cup \overline{B} \cup \overline{C}) &\text{definition of union}\\
&\equiv \overline{A} \cup \overline{B} \cup \overline{C} &\text{by meaning of set builder notation}
 \end{align*}
CarLaTeX
  • 62,716
karlu
  • 9
  • 1
    A general hint: The required arguments of (undelimited) TeX and LaTeX directives (aka "macros") must be encased in curly braces, not round parentheses. This applies to \begin as well. – Mico Oct 03 '17 at 03:49

2 Answers2

2

You need \begin{align*}, not \begin(align*):

enter image description here

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
  \overline{A \cap B \cap C}
    &\equiv x \notin A \cap B \cap C                                           && \text{definition of complement} \\
    &\equiv \neg x \in A \cap B \cap C                                         && \text{definition of does not belong symbol} \\
    &\equiv \neg(x \in A \land x \in B \land x\in C)                           && \text{definition of intersection} \\
    &\equiv \neg(x \in A) \lor \neg (x \in B) \lor \neg (x \in C)              && \text{by 1st DeMorgan's Law for logical equiv.} \\
    &\equiv x \notin A \lor x \notin B \lor x \notin C                         && \text{by definition of does not belong symbol} \\
    &\equiv x \in \overline{A} \lor x \in \overline{B} \lor x \in \overline{C} && \text{definition of complement} \\
    &\equiv x \in (\overline{A} \cup \overline{B} \cup \overline{C})           && \text{definition of union} \\
    &\equiv \overline{A} \cup \overline{B} \cup \overline{C}                   && \text{by meaning of set builder notation}
\end{align*}

\end{document}
Werner
  • 603,163
1

In addition to fixing the obvious syntax error -- it should be \begin{align*}, not \begin(align*) -- you should probably also avoid mixing expressions for sets with expressions for members of a set. In addition, the following answer uses \widebar instead of \overline to generate better-proportioned bars above the letters A, B, and C.

enter image description here

\documentclass{article}
\usepackage{amsmath} % for 'align*' env.

%% The following definition of '\widebar', for single
%% characters, is from the answer by @wipet at 
%% https://tex.stackexchange.com/a/243950, with the
%% PlainTeX code adapted for usage in LaTeX.

\newdimen\tmpdim
\def\widecharS{0.25}
\def\widebar#1{\futurelet\next\widebarA#1\widebarA}
\def\widebarA#1\widebarA{%
   \def\tmp{0}\ifcat\noexpand\next A\def\tmp{1}\fi
   \widebarE
   \ifdim\tmp pt=0pt \overline{#1}%
   \else {\mathpalette\widebarB{#1}}\fi}
\def\widebarB#1#2{%
   \setbox0=\hbox{$#1\overline{#2}$}%
   \tmpdim=\tmp\ht0 \advance\tmpdim by-.4pt
   \tmpdim=\widecharS\tmpdim
   \kern\tmpdim\overline{\kern-\tmpdim#2}}
\def\widebarC#1#2 {\ifx#1\end \else 
   \ifx#1\next\def\tmp{#2}\widebarD 
   \else\expandafter\expandafter\expandafter\widebarC
   \fi\fi}
\def\widebarD#1\end. {\fi\fi}
\def\widebarE{\widebarC A1.4 J1.2 L.6 O.8 T.5 U.7 V.3 
   W.1 Y.2 a.5 b.2 d1.1 h.5 i.5 k.5 l.3 m.4 n.4 o.6 p.4 
   r.5 t.4 v.7 w.7 x.8 y.8 \alpha1 \beta1 \gamma.6 
   \delta.8 \epsilon.8 \varepsilon.8 \zeta.6 \eta.4
   \theta.8 \vartheta.8 \iota.5 \kappa.8 \lambda.5 
   \mu1 \nu.5 \xi.7 \pi.6 \varpi.9 \rho1 \varrho1 
   \sigma.7 \varsigma.7 \tau.6 \upsilon.7 \phi1
   \varphi.6 \chi.7 \psi1 \omega.5 \cal1 \end. }

\begin{document}

\begin{align*}
\overline{A \cap B \cap C} 
& \equiv \{\,x\mid x \notin (A \cap B \cap C) \,\}
&&\text{by definition of complement} \\
& \equiv \{\,x\mid \neg x \in (A \cap B \cap C ) \,\}
&&\text{by definition of does not belong symbol} \\
& \equiv \{\,x\mid \neg(x \in A \land x \in B \land x\in C) \,\}
&&\text{by definition of intersection} \\
& \equiv \{\,x\mid \neg(x \in A) \lor \neg (x \in B) \lor \neg (x \in C) \,\}
&&\text{by 1st DeMorgan's Law for logical equiv.} \\
& \equiv \{\,x\mid (x\notin A) \lor (x\notin B) \lor (x\notin C) \,\}
&&\text{by definition of does not belong symbol} \\
& \equiv \{\,x\mid (x \in \widebar{A}) \lor (x \in \widebar{B}) \lor (x \in \widebar{C}) \,\}
&&\text{by definition of complement} \\
& \equiv \{\,x\mid x \in (\widebar{A} \cup \widebar{B} \cup \widebar{C}) \,\}
&&\text{by definition of union}\\
& \equiv \widebar{A} \cup \widebar{B} \cup \widebar{C} 
&&\text{by meaning of set builder notation}
\end{align*}

\end{document}
Mico
  • 506,678