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\centerline{Math 101}
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\centerline{Homework 0}
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\noindent 1) What is the negation of the following statement:
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$\exists \ell$ such that $\forall \varepsilon>0$, $\exists N\in \mathbb{N}$ such that if $n>N$ then $|n-\ell|<\varepsilon$.
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\noindent 2) Suppose you want to use induction to prove that the product of any finite number of odd numbers is odd. What is $P(n)$? You do not have to prove anything.
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\noindent 3) What do each of the following statements mean? Is each statement true or false? You do not have to prove anything. But write a sentence saying informally why you believe the statement is true or false.
a) $\forall x\in \mathbb{R}$, $\exists n\in \mathbb{N}$ such that $n>x$.
b) $\exists n\in \mathbb{N}$ such that $\forall x\in \mathbb{R}$ $n>x$.
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\noindent 4) Prove that the sum and product of two rational numbers is rational. Prove that the sum and product of a non-zero rational and an irrational is an irrational. (Note: you may assume that the sum and product of integers is an integer).
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\noindent 5) A function $f:\mathbb{R}\to \mathbb{R} $ is said to be {\it onto} if for
every $b \in \mathbb{R}$, there exists $a \in \mathbb{R}$ such that $f(a) = b$. A function $f : \mathbb{R} \to \mathbb{R}$ is said to be {\it one-to-one}
if $f(a_1) = f(a_2)$ implies that $a_1 = a_2$. Suppose that $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are one-to one and onto. Prove that $g\circ f$ is one-to-one and onto.
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\centerline{Homework 1}
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\noindent 1) Use induction to prove that for any natural $n >1$, and any functions $f_1, f_2, \dots, f_n$ from the reals to the reals which are one-to-one and onto, then the composition $f_n\circ f_{n-1}\circ\dots\circ f_{1}$ is one-to-one and onto. \medskip
\noindent 2) We say a function $g:\mathbb{R}\to \mathbb{R}$ is {\it invertible} if there exists a function $g^{-1}:\mathbb{R}\to \mathbb{R}$ such that $g\circ g^{-1}(x)=x$. Prove that if $f_1$, \dots, $f_n$ are invertible functions from the reals to the reals then the inverse of the function $f_1\circ\dots\circ f_n$ is the function $f_n^{-1}\circ\dots\circ f_1^{-1}$ \medskip
\noindent 3) Let $A\subseteq \mathbb{R}$. Use induction to prove that for any natural $n>1$, if $B_1$, \dots, $B_n\subseteq \mathbb{R}$ then
$$A\cap (B_1\cup\dots\cup B_n)=(A\cap B_1)\cup \dots\cup (A\cap B_n)$$
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\noindent 4) Use induction to prove that every finite set of real numbers with at least two elements has a smallest element.
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\noindent 5) Prove that the product of any finite number of odd numbers is odd. The product is only defined when you have at least two numbers.
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\centerline{Homework 2}
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\noindent 1) Let $a>0$ and let $-a0$ and suppose that $x^2b$ we have $a\leq b_1$, then $a\leq b$.
\noindent Hint: Prove the contrapositive.
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