#1. Let $P ( x )$ and $Q ( x )$ be open sentences containing the variable x, and consider the following statements.

$$
A : \forall x , ( P ( x ) \Rightarrow Q ( x ) ) \quad { \mathrm { ~ a n d ~ } } \quad B : ( \forall x , P ( x ) ) \Rightarrow ( \forall x , Q ( x ) )
$$

(a) Prove that $A \Rightarrow B ,$

(b) Give an example of open sentences $P ( x )$ and $Q ( x )$ to show that $B \Rightarrow A$ need not be true.

#2. Recall that the Fibonacci numbers are defined by $F _ { 0 } = 1 , F _ { 1 } = 1$ , and then

$$
F _ { n } = F _ { n - 1 } + F _ { n - 2 } \quad { \mathrm { f o r ~ i n t e g e r s ~ } } n \geq 2 .
$$

Prove that any two successive Fibonacci numbers $F _ { n } , F _ { n + 1 }$ are relatively prime.

$\# 3$ Prove that there is exactly one real value of x satisfying $x ^ { 3 } = 2 9 - x$

#4. (a) Give the $\epsilon , \delta$ definition for continuity of a function $f : \mathbb { R } \to \mathbb { R }$ at a point $x _ { 0 } \in \mathbb { R }$

(b) Assume that $f : \mathbb { R } \to \mathbb { R }$ is continuous and that $\left\{ a _ { n } \right\}$ is a sequence of real numbers with $\operatorname* { l i m } _ { n \to \infty } a _ { n } = L$ . Prove that $\operatorname* { l i m } _ { n \to \infty } f ( a _ { n } ) = f ( L )$

$\# 5 .$ Suppose that $T : \mathbb { R } ^ { 2 }  \mathbb { R } ^ { 2 }$ is a linear transformation satisfying $T ( 1 , 1 ) = ( 5 , 3 )$ and $T ( 2 , 3 ) = ( 7 , 9 )$ . Find the standard matrix of T.

#6. Suppose A is a $3 \times 3$ matrix and $\vec { v } _ { 1 } , \vec { v } _ { 2 } , \vec { v } _ { 3 } \in \mathbb { R } ^ { 3 }$ . Prove the following.

(a) If $\vec { v } _ { 1 } , \vec { v } _ { 2 } , \vec { v } _ { 3 }$ are linearly dependent, then $A \vec { \upsilon } _ { 1 } , A \vec { \upsilon } _ { 2 } , A \vec { \upsilon } _ { 3 }$ are linearly dependent.

() If $\vec { v } _ { 1 } , \vec { v } _ { 2 } , \vec { v } _ { 3 }$ are linearly independent and $A \vec { v } _ { 1 } , A \vec { v } _ { 2 } , A \vec { v } _ { 3 }$ are linearly dependent, then A is singular.

#7. Find all cube roots of 2-2i and express them in the standard form $a + b i$

#8. (a) Provide examples to show that the series $\sum _ { n = 1 } ^ { \infty } a _ { n } ^ { 2 }$ may or may not converge when the series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges conditionally.

(b) Prove that if the series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges absolutely, then the series $\sum _ { n = 1 } ^ { \infty } a _ { n } ^ { 2 }$ must converge.