%\input{tcilatex} \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsfonts} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2606} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %TCIDATA{Created=Thursday, July 22, 2004 13:38:11} %TCIDATA{LastRevised=Wednesday, April 04, 2007 14:47:25} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=Math.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %F=36,\PARA{038

\hfill \thepage} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section*{Problems and Comments for Section 6} \vspace{1pt} \textbf{Problems: }6.1, 6.2, 6.5, 6.8 \vspace{1pt} \textbf{Comments: }In mathematics everything should be a set. An ordered pair $(a,b)$ is not a set. It should be something different from the set consisting of $a$ and $b.$We have $\{a,b\}=\{b,a\}$ but $(a,b)\neq (b,a)$ unless $a=b.$The Kuratowski definition of an ordered pair is: $(a,b)=\{\{a\},\{a,b\}\}$ You may try to prove the following \begin{proposition} $(a,b)=(c,d)$ if and only if $a=c$ and $b=d.$ \end{proposition} Notice that in Kuratowski's definition of an ordered pair $(a,b)$, the first component is the only element of the singleton $\{a\}$ in $(a,b)$ while the second component is either also $a$ or the element $b~$in $\{a,b\}$ if $b$ is different from $a.$ \end{document}