10 Commits

Author SHA1 Message Date
venus
fe0adef332 changed proxy to traefik 2026-03-31 02:37:42 -05:00
venus
8efe0b7836 Merge branch 'master' of https://git.riverrooks.dev/venus/blog 2026-03-31 02:08:05 -05:00
venus
52e2f446d1 working on traefik integration 2026-03-31 02:07:49 -05:00
venus
7b3dce9bc8 added notes to gitignore 2026-03-11 18:34:24 -05:00
venus
bc00607685 updated index 2026-03-11 18:24:04 -05:00
venus
c03c37f691 live pull works!! 2026-03-05 02:03:16 -06:00
venus
c4634c0a0c set update to post again 2026-03-05 01:56:19 -06:00
venus
35cd103751 set update to post only 2026-03-05 01:53:19 -06:00
venus
eddd70c701 fixed live webhook update 2026-03-05 01:50:07 -06:00
venus
4d1eae0b25 fixed live pull issue 2026-03-05 01:46:02 -06:00
18 changed files with 632 additions and 55 deletions

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'
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@@ -1,28 +0,0 @@
from flask import Flask
from app import build
from pathlib import Path
import markdown
import os
from dotenv import load_dotenv
app = Flask(__name__)
PRIVATE_VAULT_DIR = Path("/vault")
PUBLIC_VAULT_DIR = "/content"
build.obsidian_vault(PRIVATE_VAULT_DIR) # initialize the private obsidian repo
build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR) # initialize the public notes from the private repo
@app.route("/")
def index():
md_content = "# Welcome to my blog!\nThis is rendered from **Markdown**.\n##[test](http://localhost/test)"
html_content = markdown.markdown(md_content)
return html_content
@app.route("/api/vault-update") #webhook for vault updated
def update_vault():
# TODO SECURE THIS WITH SECRETTTTT or auth header
print(build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR))# initialize the public notes from the private repo
return "vault-rebuilt"
@app.route ("/<filename>") # renders a filename if not otherwise specified
def render_post(filename):
return build.html_file(filename, PUBLIC_VAULT_DIR)

2
.gitignore vendored
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@@ -1,3 +1,5 @@
app/__pychache__
public-vault
compose.yml
content/
public-vault/

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@@ -17,11 +17,12 @@ build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR) # initialize the public
def index():
md_content = "# Welcome to my blog!\nThis is rendered from **Markdown**.\n##[test](http://localhost/test)"
html_content = markdown.markdown(md_content)
return html_content
@app.route("/api/vault-update") #webhook for vault updated
return build.html_file("index.md", PUBLIC_VAULT_DIR)
@app.route("/api/vault-update", methods=['POST', 'GET']) #webhook for vault updated
def update_vault():
# TODO SECURE THIS WITH SECRETTTTT or auth header
print(build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR))# initialize the public notes from the private repo
build.obsidian_vault(PRIVATE_VAULT_DIR) # initialize the private obsidian repo
build.public_vault(PRIVATE_VAULT_DIR, PUBLIC_VAULT_DIR)# initialize the public notes from the private repo
return "vault-rebuilt"
@app.route ("/<filename>") # renders a filename if not otherwise specified
def render_post(filename):

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@@ -45,6 +45,7 @@ def obsidian_vault(dest = "/vault"): # makes sure there is a vault in dest
return 1
def public_vault(privateVault = "/vault", dest = "/content"): # build the public vault in dest from an obsidian repo in src
#TODO delete old documents
vault = Vault(privateVault)
if not(vault):
print("could not find vault")

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@@ -1,13 +1,24 @@
services:
app:
container_name: blog
build:
args:
- DEBUG_MODE=1
- obsidian_vault_url=https://git.riverrooks.dev/Personal-Wiki
- obsidian_vault_token=bd8cd9301ae2c1c5bacfb3340492acb5e862686a
ports:
- '8188:80'
- obsidian_vault_token=${VAULT_TOKEN}
volumes:
- ./content:/content #public
- ./public-vault:/vault #private
labels:
- "traefik.enable=true"
- "traefik.http.routers.blog.rule=Host(`riverrooks.dev`)"
- "traefik.http.routers.blog.entrypoints=websecure"
- "traefik.http.routers.blog.tls.certresolver=myresolver"
- "traefik.http.services.blog.loadbalancer.server.port=80"
- "traefik.docker.network=traefik"
networks:
- traefik
networks:
traefik:
external: true

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---
tags:
- AI
- notes
- coding-notes
- docs
- public
---
### 1. Basic Syntax
The standard structure of an `awk` command:
Bash
```
awk [options] 'BEGIN { initialization } pattern { action } END { finalization }' file
```
commands
- **`BEGIN { ... }`**: Runs _once_ before the first line is read. Great for setting variables or printing headers.
- **`pattern`**: A condition (regex or math) that must be met for the action to run.
- **`{ action }`**: The command(s) executed on each line that matches the pattern.
- **`END { ... }`**: Runs _once_ after the last line is processed. Great for printing totals or summaries.
---
### 2. Common Command-Line Flags
|**Flag**|**Description**|**Example**|
|---|---|---|
|`-F`|Defines the field separator (default is space/tab)|`awk -F"," '{print $1}' file.csv`|
|`-v`|Assigns an external variable to use inside AWK|`awk -v max=100 '$1 > max' file`|
|`-f`|Reads AWK commands from a script file|`awk -f script.awk input.txt`|
|`-i inplace`|Edits the file in place (requires GNU awk)|`gawk -i inplace '{print $1}' file`|
---
### 3. Built-in Variables
|**Variable**|**Meaning**|**Example Use Case**|
|---|---|---|
|`$0`|The entire current line|`print $0`|
|`$1, $n`|The 1st, 2nd... _n_th field (column)|`print $2, $4`|
|`NR`|Number of Records (Current line number)|`NR > 1` (Skip headers)|
|`NF`|Number of Fields (Total columns in current line)|`print $NF` (Print the last column)|
|`FS`|Input Field Separator (Same as `-F`)|`BEGIN { FS=":" }`|
|`OFS`|Output Field Separator|`BEGIN { OFS="," }`|
|`RS`|Input Record Separator (Default: newline)|`BEGIN { RS="" }` (Paragraph mode)|
|`ORS`|Output Record Separator (Default: newline)|`BEGIN { ORS="\n\n" }` (Double space)|
---
### 4. Operators & Conditionals
**Relational & Logical Operators:**
- `==`, `!=`, `<`, `>`, `<=`, `>=` (Standard math comparisons)
- `~` (Matches regular expression)
- `!~` (Does not match regular expression)
- `&&` (AND), `||` (OR), `!` (NOT)
**Control Structures (used inside action blocks):**
```
# If/Else Statement
{ if ($1 > 50) print "Pass"; else print "Fail" }
# For Loop
{ for (i=1; i<=NF; i++) print $i }
# While Loop
{ i=1; while (i<=NF) { print $i; i++ } }
```
---
### 5. Essential Built-in Functions
|**Function**|**Description**|**Example**|
|---|---|---|
|`length(str)`|Returns the length of a string|`length($0) > 80`|
|`tolower(str)`|Converts string to lowercase|`print tolower($1)`|
|`toupper(str)`|Converts string to uppercase|`print toupper($1)`|
|`substr(s,p,l)`|Extracts substring from _s_ starting at _p_ with length _l_|`print substr($1, 1, 3)`|
|`gsub(r,s,t)`|Globally replaces regex _r_ with string _s_ in target _t_|`gsub(/apple/, "orange", $0)`|
---
### 6. Handy Everyday One-Liners
**Printing & Filtering:**
- **Print the last column of every line:**
Bash
```
awk '{ print $NF }' file.txt
```
- **Print lines longer than 80 characters:**
Bash
```
awk 'length($0) > 80' file.txt
```
- **Print lines matching "Error" or "Warning":**
Bash
```
awk '/Error|Warning/' file.log
```
**Math & Summaries:**
- **Sum the values in the first column:**
Bash
```
awk '{ sum += $1 } END { print sum }' numbers.txt
```
- **Calculate the average of the first column:**
Bash
```
awk '{ sum += $1 } END { print sum / NR }' numbers.txt
```
- **Count the number of empty lines:**
Bash
```
awk '/^$/ { count++ } END { print count }' file.txt
```
**Text Manipulation:**
- **Remove duplicate consecutive lines (simulates `uniq`):**
Bash
```
awk 'a != $0; { a = $0 }' file.txt
```
- **Number each line (simulates `cat -n`):**
Bash
```
awk '{ print NR, $0 }' file.txt
```
---

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---
tags:
- code
- notes
- docs
- School
- coding-notes
- CS-101
- public
---
# overview
OOP language
C code runs *almost* unchanged from C
```embed
title: "cplusplus.com"
image: "https://cplusplus.com/favicon.ico"
description: ""
url: "https://cplusplus.com/"
favicon: ""
```
```embed
title: "W3Schools.com"
image: "https://www.w3schools.com/images/w3schools_logo_436_2.png"
description: "W3Schools offers free online tutorials, references and exercises in all the major languages of the web. Covering popular subjects like HTML, CSS, JavaScript, Python, SQL, Java, and many, many more."
url: "https://www.w3schools.com/cpp/cpp_intro.asp"
favicon: ""
aspectRatio: "52.293577981651374"
```
```cpp
#include <iostream>
#include <stdexcept
template <typename T>
class Vector{
private:
T* data:
size_t sz;
size_t cap;
}
```
$\approx$

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#public
all non-osint CTF challenges can be solved through recursive steps until a flag is found
# 1. Explore
Analyze any data in plain text for a flag, or for the next command to run such as the results of a program.
>`cat problem.txt`
> `xxd -d asdasd238uasdkh`
> `curl challenge.org/asd`
Either solve the challenge with the flag, or determine the next program to execute
# 2 replicate
Build an isolated environment to replicate only the next program, and solve that step locally
# 3 Inject
Inject the payload that works on in the isolated environment
repeat Step 1

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@@ -1,7 +1,7 @@
#public
- **Cyber Crime**: Use computer/digital device to further illegal ends
> "nothings black and white in what we do, everything's got some gray"
## Types of crime
# Types of crime
- **Crimes Against People**: A digital crime where the victim is a person
- **Crimes Against Property**: A digital crime that damages, or illegally interacts with property
- **Crimes Against Government**: A digital crime to undermine the efficacy of a government
@@ -41,7 +41,7 @@ Nation-state attacks on power grids before Wars
- **Sneaker Net**: Moving information across barriers physicality
> proving an attack is one of the hardest questions to answer
# U.S. National cyber strategy
- **U.S. National cyber strategy**: anually updated policy document from the white house detailing the national cyber objectives
- **U.S. National cyber strategy**: annually updated policy document from the white house detailing the national cyber objectives
> it has been similar last few years
@@ -55,13 +55,14 @@ Nation-state attacks on power grids before Wars
- Shaping adversary behavior
- regulatory environment
- Federal government security
- criticial infrastructure security
- critical infrastructure security
- cyber skills workforce gap
> How can you deter people outside of your country and laws? [[attack back]]
> number of cyber jobs in US is increasing faster than the rate of new PHD grads
# Identity Theft
- **Identity Theft**: Unlawful use of another's [Personally Identifiable Information][Definitiona/Personally Identifiable Information]

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People submit week-at-a-galnce screenshots, or some form of their calendar.
Turn that into an ICS
Plugin discord bot that recognizes
>/schedule @p1 @p2
And finds available times to select. Then sends dm to users, emails them, and optional creates discord event
#public

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#public
# Work and energy
| Formula | equation | Usage |
| -------------------------------------- | ----------------------------------------------- | ----------------------------------- |
| Gravitational Potential Energy ($U_g$) | $U_g=mgh$ | Changes in height |
| Kinetic Energy | $K=\frac{1}{2}mv^2$ | Energy of motion |
| Work-Energy Theorem | $\Delta E_{mec} = \Delta K+\Delta U=W_{other}*$ | calculating the effects of friction |
| Variable | Symbol | SI Unit | Notes |
| --- | --- | --- | --- |
| Mass | m | kg | Convert from grams (g) if necessary. |
| Height / Distance | h,d | m | Measures vertical or horizontal displacement. |
| Velocity / Speed | v | m/s | Convert from km/h by dividing by 3.6.+3 |
| Gravity | g | m/s2 | Earth's standard is approximately 9.8 m/s2. |
| Energy / Work | $U_g$,K,W | J | 1 Joule = 1 kg⋅m2/s2. |
# Center Of Mass (CM)
| Formula | equation | Usage |
| -------------------------------------- | ----------------------------------------------- | ----------------------------------- |
| Position of COM X| $x_{com}=\frac{1}{M}\sum_{i=1}^{n}M_ix_i$ | find COM X|
| Position of COM Y| $y_{com}=\frac{1}{M}\sum_{i=1}^{n}M_iy_i$ | find COM Y|
| velocity of COM | $v_{com}=\frac{\sum m_iv_i}{M}$ | find the Velocity of COM |
| Variable | Symbol | SI Unit | Notes |
| --- | --- | --- | --- |
| Position (X or Y) | $x_{com},y_{com}$ | m | Coordinates on the Cartesian plane.+ 1|
| Total Mass | M | kg | The sum of all individual masses (m1+m2+…) |
| Velocity of CM | vcom | m/s | The speed at which the entire system's balance point moves |
# Linear Momentum and impulse
| Formula | equation | Usage |
| -------------------------------------- | ----------------------------------------------- | ----------------------------------- |
| Linear Momentum | p = mv | find linear momentum |
| Impulse | $J=\Delta p=F_{avg}\Delta t=\int F(t)dt$ | change in momentum. Also area under Force-time graph |
| Conservation Law | $P_{initial}=P_{final}$ | if net external force is 0, totoal momentum is constant |
| Elastic Collisions (1D) | $v_{1f}=(\frac{m_1-m_2}{m_1+m_2})v_i+(\frac{2m_2}{m_1+m_2})v_2i$| momentum and kinetic energy are conserved |
| completely inelastic collisions | $m_1v_{1i}+m_2v_{2i}=(m_1+m_2)v_f$ | find situation where the objects stick together |
| Variable | Symbol | SI Unit | Notes |
| --- | --- | --- | --- |
| Momentum | p | kg⋅m/s | Calculated as mass × velocity. |
| Impulse | J | N⋅s | Also equivalent to kg⋅m/ s.|
| Force | F | N | 1 Newton = 1 kg⋅m/s2.+2 |
| Time | t,$\Delta$t | s | The duration of the force application.+1|

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#public
# Homework
Questions (P7.)
1. a
1. A. True
2. B. Order does not matter in sets
2. MISSISSIPPI
3.
1. $\subseteq$
2. $\in$
3. $\subseteq$
4. $\in$
5. $\in$ x wrong $\emptyset$ is a $\subseteq$ of all sets
6. $\subseteq$
4. 9.
1. a) $\{S_4, S_5, S_9\}$
2. b) **??**
3. c) quadrillion
4. d)
1. F
2. T (if order does not matter)
3. T
4. F
5. T
6. T
7. F
8. F
9. F
10. T
11. F
12. F
13. T
14. F
15. T
16. T
5. 10.
1. $D_1=\{1\}, D_2=\{1,2\}, D_{10}=\{1,2,5\}$
2. b)
1. T
2. F
3. T
4. T
5. T?
6. F
7. T
8. F
9. F
10. F
11. F
12. T
3. c) $|D_{10}|=3$, $|D_{19}|=2$
4. D) $|\mathcal{D}|=9$
| Questions | Answer |
| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------- |
| 1. <br> 1. (a) True or false? {red, white, blue} = {white, blue, red}. <br> 2. (b) What is wrong with this statement: Red is the first element of the set {red, white, blue}? | a. True<br>b. Order does not matter in sets |
| 2. Which has the larger cardinality? The set of letters in the word MISSISSIPPI or the set of letters in the word FLORIDA ? | MISSIPPI |
| 3. Fill in the blank with the appropriate symbol, ∈ or ⊆. <br> 1. (a) {1, 2, 3} {1, 2, 3, 4} <br> 2. (b) 3 {1, 2, 3, 4} <br> 3. (c) {3} {1, 2, 3, 4} <br> 4. (d) {𝑎} {{𝑎}, {𝑏}, {𝑎, 𝑏}} <br> 5. (e) ∅ {{𝑎}, {𝑏}, {𝑎, 𝑏}} <br> 6. (f) {{𝑎}, {𝑏}} {{𝑎}, {𝑏}, {𝑎, 𝑏}} | |
| 9. Let 𝑆1 = {𝑜, 𝑛, 𝑒}, 𝑆2 = {𝑡, 𝑤, 𝑜}, 𝑆3 = {𝑡, , 𝑟, 𝑒, 𝑒}, and so on. <br> 1. (a) Find all 𝑘 ∈ {1, 2, . . . , 10} with \|𝑆𝑘\| = 4. <br> 2. (b) Find distinct indices 𝑗, 𝑘 with 𝑆𝑗 = 𝑆𝑘. <br> 3. (c) Find the smallest value of 𝑘 with 𝑎𝑆𝑘. <br> 4. (d) Let 𝒮 = {𝑆𝑘}40 𝑘=1. Determine whether the following statements are true or false. <br> 1. (i) 𝑆13 = {𝑛, 𝑒, 𝑖, 𝑡, , 𝑒, 𝑟} <br> 2. (ii) {𝑛, 𝑒, 𝑡} ⊆ 𝑆20 <br> 3. (iii) 𝑆1 ∈ 𝒮 <br> 4. (iv) 𝑆3 ⊆ 𝒮 <br> 5. (v) ∅ ∈ 𝒮 <br> 6. (vi) ∅ ⊂ 𝒮 <br> 7. (vii) ∅ ⊆ 𝒮 <br> 8. (viii) 𝑆1 ⊆ 𝑆11 <br> 9. (ix) 𝑆1 ⊆ 𝑆21 <br> 10. (x) 𝑆1 ⊂ 𝑆21<br> 11. (xi) {𝑛, 𝑖, 𝑒} ∈ 𝒮 <br> 12. (xii) {{𝑓, 𝑜, 𝑢, 𝑟}} ⊆ 𝒮 <br> 13. (xiii) 𝑢𝑆40 <br> 14. (xiv) 𝒫(𝑆9) ⊆ 𝒫(𝑆19) <br> 15. (xv) {𝑠, 𝑖} ∈ 𝒫(𝑆6) <br> 16. (xvi) 𝑤𝒫(𝑆2) | |
| 10. For 𝑘 ∈ {1, 2, . . . , 20}, let 𝐷𝑘 = {𝑥 𝑥 is a prime number which divides 𝑘} and let 𝒟 = {𝐷𝑘 𝑘 ∈ {1, 2, . . . , 20}}.<br> 1. (a) Find 𝐷1, 𝐷2, 𝐷10, and 𝐷20. <br> 2. (b) True or False: <br> 1. (i) 𝐷2 ⊂ 𝐷10 <br> 2. (ii) 𝐷7 ⊆ 𝐷10 <br> 3. (iii) 𝐷10 ⊂ 𝐷20 <br> 4. (iv) ∅ ∈ 𝒟 <br> 5. (v) ∅ ⊂ 𝒟 <br> 6. (vi) 5 ∈ 𝒟 <br> 7. (vii) {5} ∈ 𝒟 <br> 8. (viii) {4, 5} ∈ 𝒟 <br> 9. (ix) {{3}} ⊆ 𝒟 <br> 10. (x) 𝒫(𝐷9) ⊆ 𝒫(𝐷6) <br> 11. (xi) 𝒫({3, 4}) ⊆ 𝒟 <br> 12. (xii) {2, 3} ∈ 𝒫(𝐷12) <br> 3. (c) Find \|𝐷10\| and \|𝐷19\|. <br> 4. (d) Find \|𝒟\| | |
# Unit 1
## Krish Hw 2
1. A = {1, 2, 3}, B = {2, 3, 4} and C = {1, 2, 4}. List the elements of the specified set
1. (a) A ∩ B; | {2,3}
2. (b) A B; | {1,2,3,4}
3. (c) C\A; | {1}
4. (d) A (B ∩ C); | {2,1,3}
5. (e) (A ∩ C) (B ∩ C); | {1,2,4}
6. (f) A × B; | {(1,2),(1,3),(1,4),(2,3),(2,2),(2,4),(3,2),(3,3),(3,4)}
7. (g) B × A; | {(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2) ,(4,3)}
8. (h) (A × B) ∩ (B × A). | {(2,2),(3,3)}
2. $M_2$ = {2, 4, 6, 8, 10, · · · } and $M_3$ = {3, 6, 9, 12, 15, · · · }. Find:
1. (a) $M_2$ ∩ $M_3$;
1. $M_6$
2. (b) $M_3 \backslash M_2$
1. $\{x|x=6k-3 \space\forall\space k \in\mathbb{N}\}$

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---
tags:
- public
---
# River Rooks.dev
## Home for all of my creations ... eventually
# Check out my notes
# See [my git server](https://git.riverrooks.dev/venus) for all of my projects and dotfiles

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@@ -0,0 +1,57 @@
#public
## Malware
Malicious software:
> Software written to Do [[Harm]]
## Harm
- Devices
- Data
- Alter
- Modify
- Delete
- Corruption
# Limit Access To Information
## Privacy
Limit the amount you permit to be shared
### Healthcare
I don't want public access to my info
Choose certain info to release or not
A trip to a crisis center is private
### Personally Identifiable Information (PII)
- Info that points Directly to you
- Protect as Cyber Prof.
## CIA Triad
### Confidentiality
Keep data hidden from those who should not access it
- Encryption
- People who can see it, can't understand if there not supposed to
- Hiding Existence
Not the same as [[Malware and cybersecurity#Privacy|Privacy]]
**Keep those who don't have access from having access**
### Integrity
Ensuring that Data and System Resources are trustworthy.
Eliminate unauthorized tampering
Eliminate accidental or malicious modifications.
Examples:
Financial Records
Medical Records
Can be more important than Confidentiality.
Banking must be accurate, may be public
#### 3 Categories of integrity
##### Data Integrity
Ensuring that the data is trustworthy
##### Origin Integrity
Understanding the author and verifying the source
The author created data and it has not been changed
##### System Integrity
Securing the processes that modify the data
Eliminate Accidental, deliberate, or malicious modifications of data through systems controlling that data
### Availability
Dealing with systems and data being available for use when needed
## Secrecy
Information that should not be disclosed outside of an in-group
`Maximum Secrecy = Minimun Availability`

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@@ -9,13 +9,27 @@ flowchart TD
```
> Greatest Vulnerability is People
## Malware types
- **virus:** program to modify other programs
- **Worm**: program that spreads itself
> diff b/t virus and worm is method of movement
- **Trojan**: an innocent program that hides malware inside
- **Ransomware**: require payment to remove (often in exchange for decryption key)
- **Phishing**: Faking identity in order to build trust to encourage specific user behavior
- **DOS/DDOS**: (distributed) denial of service to overwhelm services and prevent legitimate activity from getting through
## Virus
- **Virus:** program to modify other programs
### Trojan
- **Trojan**: an innocent program that hides malware inside
looks benign and can be anything from a bad link to a malicious PDF file.
Can contain any kind of other malware such as
- key loggers
- webcams
### Examples
#### I love you
[wiki](https://en.wikipedia.org/wiki/ILOVEYOU)
- $10 billion in damages,
- 10% of worldwide competition
- attacked other computers after spreading
> at it's height, a third of the worlds computers were infected
# Threat Actors
| Group | Motivations |

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@@ -0,0 +1,157 @@
---
tags:
- Math-301
- Math
- School
- notes
- Spring-25
- public
Slides: "[[Wed 1-7.pdf]]"
Topic: Sets and Logic
Unit: 1.1 - 1.6
---
# 1.1 Basic definitions
A set is a collection of objects. The objects in a set are called its elements or members.
> Let A={a,b,c}.
> $a\in A$ -- means *a* is an element of *A*
> $d \notin A$ -- means *d* is not an element of *A*
**Def**. the [[Cardinality]] of a finite set `S`, demoted `|S|`. is the number of elements in `S`.
In the example, `|A| = 3`.
### Notation for some sets of numbers
> Natural numbers: $\mathbb{N}$ = {1,2,3,...}
> Whole Numbers: $\mathbb{W}$ = {0,1,2,3,...}
> The set of integers: $\mathbb{Z}$ = {...-3,-2,-1,0,1,2,3,} = {0,1,-1,2,-2,3,-3...}
> Rational Numbers: $\mathbb{Q}$ = $\{\frac{a}{b}\vert a,b\in\mathbb{Z},b\neq0\}$
> Set of real numbers: $\mathbb{R}$
---
## 2 subsets
- Let *A* and *B* be sets. We say that *B* is a [[subset]] of *A*, if every element of *B* is also an element of *A*, denoted $B\subseteq A$
- Two sets are [[equal]], denoted $A = B$, if $A\subseteq B$ and $B\subseteq A$
>Ex. Let *A* = $\{x\vert x\in\mathbb{Z}\hspace{7px}and\hspace{7px}0\lt x\lt6\}$.
>We see that $A=\{1,2,3,4,5\}$.
>Note: $1\in A, 4\in A$, but $6\notin A$.
>$\{1,3,5\}\subseteq A,\{2\}\subseteq A$, but $\{2\}\notin A$
>{2,4,6}$\subsetneq A$ since $6 \notin A$
>|A| = 5.
>Ex. $\mathbb{N}\subseteq \mathbb{W} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$
[[Proper subset]]: $B\subset A$ if $B\subseteq A$ and $B\neq A$
### Intervals of $\mathbb{R}$
> (a,b) = {x$\in \mathbb{R}$ | a < x < b}, where $-\infty\le a \le b \le\infty$
> \[a,b] = {x$\in \mathbb{R}$ | a $\le$ x $\le$ b}, where $-\infty\lt a \le b \lt\infty$
> \[a,b) = \{x$\in \mathbb{R}$ | a $\le$ x < b}, where $-\infty\lt a \lt b \le\infty$
> \(a,b] = \{x$\in \mathbb{R}$ | a $\lt$ x $\le$ b}, where $-\infty\le a \lt b \lt\infty$
## 3. Collections of Sets
The elements of a set may themselves be sets, and so is is a collections of sets
> Ex. $\mathcal{C}$ = { {1}, {1, 2}, {1, 2, 3} }. Note that
> {1} $\in \mathcal{C}$, {1,2} $\notin\mathcal{C}$, {1,2,3} $\in\mathcal{C}$ **??**
> {1} $\subsetneq\mathcal{C}$ since $1\notin\mathcal{C}$
> {{1}, {1,2}} $\subseteq\mathcal{C}$
> {{1}, {1,2}} $\notin\mathcal{C}$.
### Indexed Collection of Sets
Let `I` be a set. Suppose $S_i$ is a set for each `i` $\in$ `I`.
Then we say that $\{S_i\}_{i\in I}$ = $\{S_i|i\in I\}$ Is called a [[collection of sets indexed]] by `I`.
> Ex. Let $S_n$ = (n-1, n) for each $n\in\mathbb{N}$. Then
> $\{S_n\}_{n=1}^{3}$ = {$S_1,S_2,S_3$} = $\{(0,1), (1,2), (2,3)\}$
> $\{S_n\}_{n\in\mathbb{N}}$ = $\{(0,1), (1,2), (2,3),...\}$
## 4. The Empty Set
Let `E` be a set with no elements. Then for any Set `A`, we have `E`$\subseteq$`A` [[(Vacuously True)]].
If `E'` is another set with no elements, then `E`$\subseteq$`E'` `E'`$\subseteq$`E`
So `E`=`E'` Therefore, there is a unique set with no elements. We call it the [[empty set]], denoted by $\emptyset$ = {}.
**Property: For every set `A`, $\emptyset\subseteq$`A`.**
## 5. The Power Set of a Set
the [[power set]] of a set `S` is the collection of all subsets of `S` and is denoted $\wp(S)$.
> $\wp(S) = \{A | A \subseteq S\}$.
>Ex. Let $A$ = $\{a,b,c\}$. Then
>$\wp(A)\{\emptyset\,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$
>Note: |$\wp(A)$ = 8 = $2^3$ = $2^{|A|}$.
## 6. Summarizing example
Consider the set:
> $S = \{1,2,\emptyset,\{a,b\}\}$. Then
> $2\in S, 2\subsetneq S$
> ${2}\notin S, 2\subseteq S$
> > $\emptyset, 2\subsetneq S$
# 1.2 Set Operations
## 1 Intersections and Unions
>Let `A` and `B` be sets. The [[Intersection]] of `A` and `B` is the set.
>$A\cap B$ {X | x $\in$ A and x $\in$ B}
>![[Pasted image 20260114205835.png]]
The [[Union]] of `A` and `B` is the set
> $A \cup B$ = {x | $\in$ a A or x $\in$ B } ## "Or" is the "Inclusive or"
> ![[Pasted image 20260114210548.png]]
> Ex. Let `A`={0,2,4,6,8}, `B`={0,3,6,9}, and C={1,2,3,4,5,6,8,9}. Then
> (a) $A\cap B$ = {0,6}
> (b) $A\cup B = \{0,2,4,6,8,3,9\}=\{0,2,3,4,6,8,9\}$
> (c) $B\cap C$ = {3}
> (d) $A\cup(B\cap C) = \{0,2,3,4,6,8\}$
> (e) $(A\cup B)\cap C = \{2,3,4\}$
> Note: $A\cup(B\cap C)\neq(A\cup B)\cap C$
**Def. We say that two sets A and B and [[disjoint]] if $A\cap B=\emptyset$**
![[Pasted image 20260115174934.png]]
> **Theorem. Let `A` and `B` be finite sets.**
> (a) $|A\cup B|=|A|+|B|-|A\cup B|$
> ![[Pasted image 20260115175142.png]]
> (b) if `A` and `B` and disjoint then $|A\cup B| = |A| + |B|$
> ![[Pasted image 20260115175132.png]]
> (c) If $A\subseteq B, then |A|\leq|B|$
## 2 Arbitrary Collections
Let $\mathcal{C}=\{S_i|i\in I\}$ be a collection of sets indexed by a set `I` (Assume $I\neq0$.) Then the [[Intersection]] of $\mathcal{C}$ is defined as
> $\cap\mathcal{C}=\cap_{i\in I} S_i=\{x|x\in S_i$ for all $i\in I\}$.
The [[Union]] of the collection $\mathcal{C}$ is the set
>$\cup\mathcal{C}=$ $\cup_{i\in I}S_i=\{x|x\in S_i$ for at least one $i\in I\}$
For a finite collection of sets indeed by $I=\{1,2 ... , n\}$
we often write the intersection and unions of $\mathcal{C}=\{S_1,S_2, ... ,S_n\}$ as
> $\cap_{i=1}^n S_i = S_1\cap S_2\cap S_3\cap ... \cap S_n$
> $\cup_{i\in I}^n S_i=S_1\cup S_2\cup ... \cup S_n$
Def. Let $\{\S_i|i\in I\}$ be an indexed collection of sets.
> (a) The collection is [[Mutually disjoint]] if for all $i,j\in I$, if $S_i\neq S_j, then S_I =\cap S_j\space\emptyset$
> Equivalently: for all $i,j\in I$, $S_i=S_j\space or\space S_i\cap S_j=\emptyset$
> (b) The collection is [[nested]] if for all $i,j\in I$, $S_i\subseteq S_j$ or $S_j\subseteq S_i$
$S=\{x\in\mathbb{W}|x\notin M_{5}, x\in M_3\}$
---
> $B_n$ = $\{x\in\mathbb{R}\space\vert\space |x| < n\}$
$B_1$ = (-1,1)
$B_2$ = (-2,2)
$B_3$ =(-3, 3)
> [!note] incluse vs exclusion
$[-1,1]\neq (-1,1)$ where $n=\mathbb{R}$
let $A$ = $(-1,1)$
let $B$ = $[-1,1]$
$A\subseteq B$
$A\neq B$
| Q | P | A |
| ------------------ | ---------------------- | ------------------ |
| Mutually disjoint? | $B_n\subseteq B_{n+1}$ | False |
| Nested?<br> | $B_n\subseteq B_{n+1}$ | True |
| Intersect | $B\cap B_n$ | $B_1$ |
| Union<br> | $B\cup B_n$ | $(-\infty,\infty)$ |

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public: "true"
tags:
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# This is a test
and this is p
[https://localhost/test]()
[asd](https://localhost/test)