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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 $-a<x<a$.  Prove that $x^2<a^2$.   \medskip
  
\noindent   2) Let $a>0$ and suppose that $x^2<a^2$.  Prove that $-a<x<a$.  (Note you cannot take the square root of both sides of an inequality.)
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\noindent 3) Use induction to prove that every $n \in {\bf N}$ has the form $3k$, $3k+1$, or $3k+2$ for some non-negative integer $k$.
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\noindent 4) Prove that $\sqrt{3}$  is irrational using a method analogous to the one I used to show $\sqrt{2}$ is irrational.  (You cannot assume that if $3\:|\: p^2$ then $3\:|\: p$.)





 




  
 

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 \centerline{Homework 3}


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  \noindent Problems 3.4, 3.6, 3.7 in Ross.  Recall that among our axioms about the reals we assume that $0$ is not the only real number.
  
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 \noindent 4)  Let $a$ and $b\in \mathbb{R}$.  Prove that if $\forall b_1>b$ we have $a\leq b_1$, then $a\leq b$.  
 
 \noindent Hint:  Prove the contrapositive.
  

  

 
 


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