Attempt all problems; they are weighted equally.
Write N for the set of positive integers and R for the set of real numbers.

1. Write the negations of these sentences in as "smooth" a way as possible.
   (In particular, you may not simply append $\mathfrak { a } _ { \mathbf { I t } }$ is not the case that ...". Also, you should make explicit any "hidden" quantifiers.)

(a) There is a real number x such that for every real number $y , | x - y | > 1$

(b) A real-valued function that is continuous on a closed interval attains a minimum value on that interval.

(c) $3 n + 1$ is even if and only if $n ^ { 2 } + 4$ is prime.

2. Let $A = { \binom { 7 } { 1 } }$ . Find an invertible matrix P and a diagonal matrix D with $P ^ { - 1 } A P = D$ [You should not have to compute $P ^ { - 1 } . ]$

3. Let A be an m x n real matrix.
   Write $A ^ { t }$ for the transpose of A, and $N ( A )$ for the nullspace of A. Prove that $N ( A ^ { t } A ) = N ( A )$

4. Suppose $\pmb { f } : \mathbb { R }  \mathbb { R }$ satisfies $f ( x y ) = x f ( y ) + y f ( x )$ for all ${ \pmb x } , { \pmb y } \in \mathbb { R }$ Prove that $f ( 1 ) = 0$ and that $f ( u ^ { n } ) = n u ^ { n - 1 } f ( u )$ for all $\pmb { n } \in \mathbb { N }$ and ${ \pmb u } \in \mathbb { R }$

Give an e-δ proof that lim ${ \frac { 1 } { x ^ { 2 } + 1 } } = { \frac { 1 } { 5 } }$

6. Let $f : X \to Y$ be a (not necessarily invertible!)
   function, and $A \subseteq X$

(a) Prove that $A \subseteq f ^ { - 1 } ( f ( A ) )$

(b) Prove that if f is injective (one-to-one) then $A = f ^ { - 1 } ( f ( A ) )$

(c) Give an example for which $A \neq f ^ { - 1 } ( f ( A ) )$

7. Prove that the line integral $\int _ { C } ( x + y ^ { 3 } ) d x + ( e ^ { y } + 3 x y ^ { 2 } )$ dy is path-independent; i.e., it depends only on the endpoints of C.

8. Let $f _ { n } ( x ) = { \frac { n x } { n + x } }$ for ${ \pmb x } \in [ { \bf 0 } , \infty )$ and $\pmb { n } \in \mathbb { N }$

(a) Find a function f such that $\{ f _ { n } \}$ converges to f pointwise on $| 0 , \infty )$

(b) Is the convergence uniform on $\lbrack 0 , \infty ) \ ?$ Justify your answer.

9. Suppose R is a commutative ring (with 1), I is a proper ideal in $R ,$ and $a \in R$ Suppose $\langle a \rangle + I = R$ Prove that $a + I$ s a unit ., invertible) i the quotint ing $R / I$

(For half credit: Prove that if a and n are relatively prime integers, then $a + n \mathbb { Z }$ is a unit in $\mathbb { Z } / { \mathfrak { n } } \mathbb { Z } . )$