Homework #2

EE 503: Fall 2025

Assignment Details

Assigned: 02 September
Due: Tuesday, 09 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

Use truth tables to prove whether these propositional assertions are valid or invalid:

$[(P \wedge Q) \Rightarrow R] \Rightarrow [(P \wedge Q) \Rightarrow [(P \wedge Q) \wedge R]]$.
$(P \Rightarrow R) \Rightarrow [(Q \rightarrow R) \rightarrow ((P \vee Q) \rightarrow R)]$.
$[P \rightarrow (Q \rightarrow R)] \Leftrightarrow [(P \wedge Q) \Rightarrow R]$.
$[\sim (P \Rightarrow Q) \Rightarrow R] \Rightarrow [(\sim P \vee Q) \vee R]$.

Problem 2

Use mathematical induction to prove these theorems for all positive integers $n \ge 1$:

$\displaystyle \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$.
$23^n - 1$ is divisible by 11.
$(2n)! > 2^n (n!)^2$.

Problem 3

Prove or disprove:

$2^A \cap 2^B = 2^{A \cap B}$.
$2^{A^C} = (2^A)^C$.
$\cup_\alpha 2^{A_\alpha} = 2^{\cup_{\alpha} A_\alpha}$ for arbitrary set collection $\{A_\alpha\}$.

Problem 4

Let $\Omega = \{a, b, c, d, e, f\}$. Find the sigma-algebra $\sigma(\emptyset)$ that the empty set $\emptyset$ generates. Find the sigma-algebra $\sigma(\{b, e\})$. Find the sigma-algebra $\sigma(\{b, e, f\})$. Find $\sigma(\sigma(\{b\}) \cup \sigma(\{f\}))$.

Problem 5

We do not know how the Ancient Green engineer Archimedes proved his famous result $\frac{265}{153} < \sqrt{3} < \frac{1351}{780}$ described in “Measurement of a Circle”. Use a proof by contradiction to show that $\sqrt{3}$ is an irrational number.

Problem 6

Prove by induction that the $n$ eigenvectors $e_1, \ldots, e_n$ of the $n$-by-$n$ matrix $A$ are linearly independent if the $n$ corresponding eigenvalues $\lambda_1, \ldots, \lambda_n$ are distinct. The column vector $e_k$ is an eigenvector of $A$ with scalar eigenvalue $\lambda_k$ if and only if $A e_k = \lambda_k e_k$ and $e_k$ is not the null vector. The $p$ column vectors $v_1, \ldots, v_p$ are linearly independent if and only if the following implication holds for all complex scalars $c_1, \ldots, c_p$: $\sum_{k=1}^{p} c_k v_k = 0$ implies that all the scalars $c_k$ are zero: $c_1 = \cdots = c_p = 0$.

--- title: "Homework #2" 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:** 02 September **Due:** Tuesday, 09 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 Use truth tables to prove whether these propositional assertions are valid or invalid: a. $[(P \wedge Q) \Rightarrow R] \Rightarrow [(P \wedge Q) \Rightarrow [(P \wedge Q) \wedge R]]$. b. $(P \Rightarrow R) \Rightarrow [(Q \rightarrow R) \rightarrow ((P \vee Q) \rightarrow R)]$. c. $[P \rightarrow (Q \rightarrow R)] \Leftrightarrow [(P \wedge Q) \Rightarrow R]$. d. $[\sim (P \Rightarrow Q) \Rightarrow R] \Rightarrow [(\sim P \vee Q) \vee R]$. ## Problem 2 Use mathematical induction to prove these theorems for all positive integers $n \ge 1$: a. $\displaystyle \sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$. b. $23^n - 1$ is divisible by 11. c. $(2n)! > 2^n (n!)^2$. ## Problem 3 Prove or disprove: a. $2^A \cap 2^B = 2^{A \cap B}$. b. $2^{A^C} = (2^A)^C$. c. $\cup_\alpha 2^{A_\alpha} = 2^{\cup_{\alpha} A_\alpha}$ for arbitrary set collection $\{A_\alpha\}$. ## Problem 4 Let $\Omega = \{a, b, c, d, e, f\}$. Find the sigma-algebra $\sigma(\emptyset)$ that the empty set $\emptyset$ generates. Find the sigma-algebra $\sigma(\{b, e\})$. Find the sigma-algebra $\sigma(\{b, e, f\})$. Find $\sigma(\sigma(\{b\}) \cup \sigma(\{f\}))$. ## Problem 5 We do not know how the Ancient Green engineer Archimedes proved his famous result $\frac{265}{153} < \sqrt{3} < \frac{1351}{780}$ described in "Measurement of a Circle". Use a proof by contradiction to show that $\sqrt{3}$ is an irrational number. ## Problem 6 Prove by induction that the $n$ eigenvectors $e_1, \ldots, e_n$ of the $n$-by-$n$ matrix $A$ are linearly independent if the $n$ corresponding eigenvalues $\lambda_1, \ldots, \lambda_n$ are distinct. The column vector $e_k$ is an *eigenvector* of $A$ with scalar *eigenvalue* $\lambda_k$ if and only if $A e_k = \lambda_k e_k$ and $e_k$ is not the null vector. The $p$ column vectors $v_1, \ldots, v_p$ are *linearly independent* if and only if the following implication holds for all complex scalars $c_1, \ldots, c_p$: $\sum_{k=1}^{p} c_k v_k = 0$ implies that all the scalars $c_k$ are zero: $c_1 = \cdots = c_p = 0$.