\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}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section{Problems and Comments on Section 1} \textbf{Problems: }1.3, 1.9 For the time being, you can skip section 5. This example will be better understood after we have covered rings (Section 16, Exercises 16.16, 16.17): For any set $X$ one has that the powerset $P(X)$ of $X$ is a \textit{boolean ring} where addition is the symmetric difference: $+=$ $\bigtriangledown $ and multiplication is the intersection: $\cdot =\cap $, zero is the empty set: $0=\emptyset $ and one is the whole set \ $1=X$. \textbf{Comments:} The text defines only \textit{binary }operations\textit{. }An \textit{n-ary operation} is a map \[ f:A^{n}\rightarrow A \] that is for every \textit{n-tuple} of elements from $A$ a unique element $% f(a_{1},a_{2},\ldots ,a_{n})\in A$ is assigned as operation value for $f.$A \textit{unary operation }assignes to every element of $A$ a uniquely determined element of $A$. That is, a unary operation is just a map on $A$. Taking the additive inverse of an integer is an example of a unary operation:% \[ -:% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion \rightarrow %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion ,n\longmapsto -n \] Now, what is a \textit{nullary operation?} Well, we have to say what $A^{0}$ should be. It is meaningful to define $A^{0}$ as $1.$Recall that $0$ is defined as the empty set, $0=\emptyset $ and $1$ as the set which contains only zero, that is the empy set: $1=\{\emptyset \}.$ Hence,% \[ A^{0}=\{0\} \] and a nullary operation then is a map \[ c:A^{0}\rightarrow A,0\mapsto a \] which assigns to $0$ a unique element in $A$. Such a map is called a\textit{% \ constant}. We can also think that an \textit{n-ary} operation depends on \textit{n-many }arguments. A \textit{nullary} operation depends on \textit{% zero many} arguments, it is constant. Examples for constants are: The \textit{zero} in the additive structure of the integers. For this algebra we write: \[ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion =(Z,+,-,0) \] making a notational distinction between the set $Z$ of integers and the algebraic structure on that set. This structure makes $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $ to what we will learn as a \textit{group}. A \textit{universal algebra} is a set $A$ together with a family $f_{i}$ of operations, each of an arity $n_{i}:$ \[ \mathbf{A}=(A,(f_{i})_{i\in I}) \] For the integers we can choose $I=\{0,1,2\}$ and $f_{0}=+,\ f_{1}=-,\ f_{2}=0.$Of course, we will soon extend the algebra of integers to include also multiplication. The multiplicative unit is the constant $1$: \[ %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion =(Z,+,-,0,\cdot ,1) \] is what is called the \textit{ring} of integers with addition and multiplication. \end{document}