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 ... ... @@ -53,7 +53,8 @@ or the positive integers $\Z^+=\set{1,2,\ldots}$. and~$k$ is the \definend{quotient}. (Alternative wordings are: $d$~\definend{is a factor of}~$n$, or $d$~\definend{goes evenly into}~$n$, $d$~\definend{goes evenly into}~$n$, $n$~\definend{is divisible by}~$d$, or $n$~\definend{is a multiple of}~$d$.) We denote the relationship as $d\divides n$ if $d$~is a divisor of~$n$ ... ... @@ -63,8 +64,8 @@ or the positive integers $\Z^+=\set{1,2,\ldots}$. \begin{df} A number is \definend{even} if it is divisible by~$2$, otherwise it is \definend{odd}. (We may instead say that a number has \definend{even parity} or \definend{odd parity}.) (We may instead say that the \definend{parity} is even or odd.) \end{df} The notation $d\divides n$ signifies a relationship between two integers. ... ... @@ -2074,7 +2075,7 @@ An object~$x$ that belongs to a set~$A$ is an \definend{element} or \definend{member} of that set, denoted $x\in A$. Denote that $x$ is not an element with~$x\notin A$. Two sets are equal if and only if they have the same elements. Sets are equal if and only if they have the same elements. % \end{df} %\noindent (Read $\in$' as is an element of'' rather than `in'' to avoid ... ... @@ -2194,7 +2195,7 @@ Prove. \end{problem} \begin{problem} \label{ex:PropertiesOfSubset} Prove, for any sets $A$, $B$, and~$C$. Prove, for sets $A$, $B$, and~$C$. \begin{exes} \begin{exercise} \notetext{Mutual Inclusion} If $A\subseteq B$ and $B\subseteq A$ then $A=B$. ... ... @@ -2620,7 +2621,7 @@ where $X$ is the universe, $\universalset=X$. % \end{answer} % \end{problem} \begin{problem} \pord \begin{problem} \pord. \begin{exes} % \begin{exercise} % $A-B\subseteq A$ ... ... @@ -3554,16 +3555,17 @@ This is asymmetric because the definition puts no such condition on output elements. \begin{df} A function is \definend{one-to-one}, or \definend{$1$-$1$}, A function $\map{f}{D}{C}$ is \definend{one-to-one}, or \definend{$1$-$1$}, or an \definend{injection}, if for each value there is at most one associated argument, that is, if $f(d_0)=f(d_1)$ implies that $d_0=d_1$ for all elements $d_0,d_1$ of the domain. A function is \definend{onto}, or a \definend{surjection} if for each value there is at least one associated argument, that is, if for each element $c$ of the codomain there exists an element $d$ of the domain such that $f(d)=c$. A function that is both one-to-one and onto, so that for every value there one associated argument, that is, if $f(d_0)=f(d_1)$ for $d_0,d_1\in D$ implies that $d_0=d_1$. The function is \definend{onto}, or a \definend{surjection}, if for each member of the codomain $c\in C$ there is at least one associated argument, at least one $d\in D$ such that $f(d)=c$. A function that is both one-to-one and onto, so that for every member of the codomain there is exactly one associated argument, is a \definend{correspondence} or \definend{bijection}. \end{df} ... ...
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 ... ... @@ -6,18 +6,19 @@ Its only prerequisite is high school mathematics. \smallskip \noindent\textsc{Approach.} This course is inquiry-based (sometimes called discovery method or Moore method). This course is Inquiry-Based (sometimes called the Discovery Method or Moore Method). Students get a sequence of exercises along with definitions and a few remarks. They work through the material together, by proving statements or providing examples. This means that each person must grapple directly with the mathematics\Dash the instructor only lightly guides and the students pledge not to use outside sources\Dash talking out misunderstandings, sometimes stumbling in the dark, and sometimes having beautiful flashes of insight. They attempt these outside of class, and then in class propose solutions. The instructor only lightly guides and the students pledge not to use outside sources so they together talk through misunderstandings, sometimes stumble in the dark, and sometimes have beautiful flashes of insight. Most important is that each person must grapple directly with the mathematics. For these students, with this material, this is the best way to develop mathematical maturity. Besides, it is a great deal of fun. ... ... @@ -47,7 +48,7 @@ intellectual habits that we established at the start. \noindent\textsc{Exercises.} Where possible, nearby exercises have about the same difficulty. This gradually rises. This level gradually rises. Some exercises have multiple items; these come in two types. If the items are labeled \textsc{A}, \textsc{B}, etc., ... ...