\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*}
答案1
你\begin{align*}不需要\begin(align*)：
\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}
答案2
除了修复明显的语法错误——它应该是\begin{align*}，而不是\begin(align*)）之外，您还应该避免将集合表达式与成员的集合。此外，以下答案使用\widebar而不是\overline来在字母A、B和上方生成比例更好的条C。
\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}