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Commits (1)
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 ... ... @@ -290,7 +290,7 @@ possible if $j_0=j_1$). \begin{exes} \begin{exercise} $\powerset(\N)$ \ \hint let $\map{f}{\powerset(\N)}{\N}$ \ \hint suppose that $\map{f}{\powerset(\N)}{\N}$ and consider $\setbuilder{n\in\N}{n\notin f(n)}$. \end{exercise} \begin{answer} ... ... @@ -337,41 +337,4 @@ possible if $j_0=j_1$). element~$a\in\N$, and the two image numbers then differ in the $a$-th place. \end{answer} \end{exes} % \begin{exes} % \item $\powerset(\N)$ % \item $\R$ % \end{exes} % \begin{ans} % \begin{exes} % \item We will show that no $\map{f}{\N}{\powerset(\N)}$ is onto. % Consider $R=\setbuilder{n\in\N}{n\notin f(n)}$. % Clearly $R\subseteq \N$ and so $R\in\powerset(\N)$. % We will show that there is no $i\in\N$ such that $R=f(i)$. % For contradiction assume that $R=f(i)$. % Consider whether $i\in R$. % If $R=f(i)$ and $i\in R$ then that contradicts the definition of $R$ as the % collection of integers~$n$ such that $n\notin f(n)$. % The other $R=f(i)$ possibility is that $i\notin R$, which % is also a contradiction % because then $i$~satisfies the defining criteria for membership in~$R$, namely % that $i\notin R$, and therefore $i\in R$. % Either $R=f(i)$ case gives a contradiction and so $R\neq f(i)$. % \item We will show that there is a one-to-one map~$\map{f}{\powerset(\N)}{\R}$. % If $\R$ were countable\Dash if there were a one-to-one % function $\map{g}{\R}{\N}$\Dash then % that would give a one-to-one function~$\composed{g}{f}$ % from $\powerset(\N)$ to~$\N$, % contradicting the prior item. % We define the action of~$f$ on a set~$A\in\powerset(\N)$ by % giving an associated real number between $0$ and~$1$. % We will give this number's binary expansion, instead of its more usual % decimal expansion. % Let this number be such that its $i$-th binary place is $\charfcn{A}(i)$, % that is, the $i$-th binary place is $1$ if $i\in A$ and is $0$ otherwise. % This function is one-to-one because two unequal sets differ in an % element~$a\in\N$, and the two image numbers then differ in the $a$-th place. % \end{exes} % \end{ans} \end{problem}
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 ... ... @@ -30,7 +30,7 @@ We cover sets, functions and relations, and elementary number theory. We start with number theory instead of sets for the same reason that the baseball team's annual practice starts by tossing the ball and not by reading the rulebook. not by reading the rule book. Math majors take readily to proving things about divisibility and primes, whereas weeks of preliminary material is less of a lure. ... ... @@ -56,7 +56,7 @@ If the labels are (i), (ii), etc., then they together make a single assignment. I have students put proposed answers on the board for discussion, and if the items are labelled alphabetically then I ask a different student if the items are labeled alphabetically then I ask a different student to do each one while for the others I ask a single student to do them all. % This text comes in versions that differ in the number of exercises, ... ... @@ -78,7 +78,7 @@ to do each one while for the others I ask a single student to do them all. You can always use a prior result. You can also use the rules of high school algebra such as associativity of addition $x+(y+z)=(x+y)+z$, or distributivity of multiplication over addtion $x\cdot(y+z)=xy+xz$, or that a positive times a positive over addition $x\cdot(y+z)=xy+xz$, or that a positive times a positive equals a positive. Finally, you can use elementary logic, such as that a statement like $P$ and~$Q$'' is true if and only if ... ...