# Department of Mathematics PRELIMINARY EXAMINATION

Work all of the following problems, justifying your answers.
The problems are weighted evenly.

R denotes the set of real numbers and $\mathcal { P } _ { n }$ denotes the vector space of (real) polynomials of degree $\leq n .$ As usual, ${ \binom { n } { k } } = { \frac { n ! } { k ! ( n - k ) ! } }$ is the binomial coefficient.

1. a. Suppose A and B are false statements.
   Is the statement

$$
( A \implies B ) \implies ( A \lor B )
$$

true or false?
$( ^ { \ast } \lor ^ { \prime \prime }$ denotes "or.")

b.Give the negation of the statement

If all blockoids are split and some blockoid is nontrivial, then there is a short blockoid.

2.Let k be a nonnegative integer.
Prove by mathematical induction that for all $n \geq k$ we have

$$
\sum _ { m = k } ^ { n } { \binom { m } { k } } = { \binom { n + 1 } { k + 1 } } .
$$

3. Suppose $A , B ,$ and C are sets, and $f \colon B \to C$ and $g \colon A  B$ are functions.

aProve that if $f$ and g are surjective (onto), then so is $f \circ g$

b.Prove or give a counterexample: If g is not surjective, then $f \circ g$ is not surjective.

a. Complete the ${ \delta - } \varepsilon$ definition of a limit: Given a function $f : \mathbb { R } \to \mathbb { R }$ and a, $L \in \mathbb { R } ,$

$$
\operatorname* { l i m } _ { x \to a } f ( x ) = L \qquad { \mathrm { m e a n s } } \qquad \cdot \cdot \cdot
$$

Using the definition, prove that

$$
\operatorname* { l i m } _ { x \to 1 } { \frac { 1 } { 1 + x ^ { 2 } } } = { \frac { 1 } { 2 } } .
$$

5. Let $\Delta$ be the triangle in $\mathbb { R } ^ { 2 }$ with vertices (0, 1), (2, 0), and (2, 1), traversed counterclockwise.
   Evaluate the line integral

$$
\int _ { \Delta } { y d x } + 2 x d y .
$$

Suppose V and W are vector spaces and $T \colon V \to W$ is a linear transformation.
Suppose that $T ( v ) =$ 0 only when $v = 0$ Prove that if $\{ v _ { 1 } , \ldots , v _ { k } \}$ is a linearly independent set of vectors in V, then $\{ T ( v _ { 1 } ) , . . . , T ( v _ { k } ) \}$ is a linearly independent set of vectors in W.

a iu o tol:

a. An integrable function $f : \mathbb { R } \to \mathbb { R }$ so that the function $F ( x ) = \int _ { 0 } ^ { x } f ( t )$ dt is differentiable everywhere but at x = 1.

b. A sequence $\left\{ a _ { n } \right\}$ of real numbers such that $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges and $\sum _ { n = 1 } ^ { \infty } a _ { n } ^ { 2 }$ diverges.

A basis for the subspace of $\mathcal { P } _ { 3 }$ spanned by $x ^ { 2 } + x + 1 , x ^ { 3 } - x + 2 , x ^ { 3 } + x ^ { 2 } + 3$ ,and $- x ^ { 3 } + x ^ { 2 } + 2 x + 1$