Homework #3

EE 503: Fall 2025

Assignment Details

Assigned: 09 September
Due: Tuesday, 16 September at 12:00

BrightSpace Assignment: Submit your solutions to BrightSpace

Instructions

Write your solutions to these homework problems. Submit your work to Brightspace by the due date. Show all work and box answers where appropriate. Do not guess.

Problem 1

A function $f: X \rightarrow Y$ is onto (or surjective) iff for each “image” element $y \in Y$ there is a “pre-image” element $x \in X$ such that $y = f(x)$. A function $f$ is one-to-one (or injective) iff distinct pre-images have distinct images: $f(x_1) \ne f(x_2)$ if $x_1 \ne x_2$ for all $x_1 \in X$ and all $x_2 \in X$. Note that the contrapositive of the last statement states that $f(x_1) = f(x_2)$ only if $x_1 = x_2$ for all $x_1 \in X$ and all $x_2 \in X$. A function is bijective iff it is both injective and surjective (precisely when the inverse point function $f^{-1}$ exists). Suppose $A \subset X$ and $B \subset Y$ for $f: X \rightarrow Y$. Then prove or disprove:

$f(f^{-1}\{B\}) \subset B$.
$f(f^{-1}\{B\}) = B$ if $f$ is surjective.
$A \subset f^{-1}\{f(A)\}$.
$A = f^{-1}\{f(A)\}$ if $f$ is injective.
$f: X \rightarrow Y$ is bijective implies $f:2^X \rightarrow 2^Y$ is bijective.

Problem 2

Use the ratio test to determine whether the following infinite series diverge or converge:

$\displaystyle \sum_{n=1}^{\infty} \frac{n^2 2^{n+1}}{3^n}$.
$\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{2^n}{n!}$.
$\displaystyle \sum_{n=1}^{\infty} \frac{n!}{\ln(n+1)}$.
$\displaystyle \sum_{n=1}^{\infty} (\ln 2)^{-n} n^{10}$.

Problem 3

Find the interval of convergence for these power series (check both endpoints):

$\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{\ln(1 + n)}$.
$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1} (x-5)^n}{n 5^n}$.
$\displaystyle \sum_{n=1}^{\infty} \frac{n (x-2)^n}{2^n (3n - 1)}$.

Problem 4

Use the $\epsilon$-definition (i.e. garden hose) to evaluate the limit of these sequences. Given $\epsilon = 10^{-6}$ what is the smallest index $n_0$ such that $|a_n - L| < \epsilon$ for all $n \ge n_0$?

$\displaystyle \frac{1 - n^2 + n^3}{n^3 - 1}$.
$\displaystyle \frac{\sin n - \cos n}{n}$.

Problem 5

Let $A \times B = \{(x,y): x \in A \text{ and } y \in B\}$. Suppose $A = \{a_1, a_2, a_3\}$ and $B = \{b_1, b_2\}$. Then what is the Cartesian product $A \times B$? How many elements in $2^{A \times B}$? Produce four sub-collections $\mathcal{A} \subset 2^{A \times B}$ that are sigma-algebras.

Problem 6

Prove or disprove:

If $A \subset X$ and $B \subset Y$ then $A \times B \subset X \times Y$.
$(A \cup B) \times C = (A \cup C) \times (B \cup C)$.

--- title: "Homework #3" subtitle: "EE 503: Fall 2025" format: html: toc: false number-sections: false code-fold: show code-tools: true theme: cosmo --- ::: {.callout-important} ## Assignment Details **Assigned:** 09 September **Due:** Tuesday, 16 September at 12:00 **BrightSpace Assignment:** Submit your solutions to BrightSpace ::: ::: {.callout-warning} ## Instructions Write your solutions to these homework problems. Submit your work to Brightspace by the due date. Show all work and box answers where appropriate. Do not guess. ::: --- ## Problem 1 A function $f: X \rightarrow Y$ is *onto* (or *surjective*) iff for each "image" element $y \in Y$ there is a "pre-image" element $x \in X$ such that $y = f(x)$. A function $f$ is *one-to-one* (or *injective*) iff distinct pre-images have distinct images: $f(x_1) \ne f(x_2)$ if $x_1 \ne x_2$ for all $x_1 \in X$ and all $x_2 \in X$. Note that the contrapositive of the last statement states that $f(x_1) = f(x_2)$ only if $x_1 = x_2$ for all $x_1 \in X$ and all $x_2 \in X$. A function is *bijective* iff it is both injective and surjective (precisely when the inverse *point function* $f^{-1}$ exists). Suppose $A \subset X$ and $B \subset Y$ for $f: X \rightarrow Y$. Then prove or disprove: a. $f(f^{-1}\{B\}) \subset B$. b. $f(f^{-1}\{B\}) = B$ if $f$ is surjective. c. $A \subset f^{-1}\{f(A)\}$. d. $A = f^{-1}\{f(A)\}$ if $f$ is injective. e. $f: X \rightarrow Y$ is bijective implies $f:2^X \rightarrow 2^Y$ is bijective. ## Problem 2 Use the ratio test to determine whether the following infinite series diverge or converge: a. $\displaystyle \sum_{n=1}^{\infty} \frac{n^2 2^{n+1}}{3^n}$. b. $\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{2^n}{n!}$. c. $\displaystyle \sum_{n=1}^{\infty} \frac{n!}{\ln(n+1)}$. d. $\displaystyle \sum_{n=1}^{\infty} (\ln 2)^{-n} n^{10}$. ## Problem 3 Find the interval of convergence for these power series (check both endpoints): a. $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{\ln(1 + n)}$. b. $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1} (x-5)^n}{n 5^n}$. c. $\displaystyle \sum_{n=1}^{\infty} \frac{n (x-2)^n}{2^n (3n - 1)}$. ## Problem 4 Use the $\epsilon$-definition (*i.e.* *garden hose*) to evaluate the limit of these sequences. Given $\epsilon = 10^{-6}$ what is the smallest index $n_0$ such that $|a_n - L| < \epsilon$ for all $n \ge n_0$? a. $\displaystyle \frac{1 - n^2 + n^3}{n^3 - 1}$. b. $\displaystyle \frac{\sin n - \cos n}{n}$. ## Problem 5 Let $A \times B = \{(x,y): x \in A \text{ and } y \in B\}$. Suppose $A = \{a_1, a_2, a_3\}$ and $B = \{b_1, b_2\}$. Then what is the Cartesian product $A \times B$? How many elements in $2^{A \times B}$? Produce four sub-collections $\mathcal{A} \subset 2^{A \times B}$ that are sigma-algebras. ## Problem 6 Prove or disprove: a. If $A \subset X$ and $B \subset Y$ then $A \times B \subset X \times Y$. b. $(A \cup B) \times C = (A \cup C) \times (B \cup C)$.