<h1 id="union">Union<a aria-hidden="true" class="anchor-heading icon-link" href="#union"></a></h1>
<div class="portal-container">
<div class="portal-head">
<div class="portal-backlink">
<div class="portal-title">From Union</div>
<a href="/notes/muqw0evhz5f2lpn8if6f1vd" class="portal-arrow">Go to text →</a>
</div>
</div>
<div id="portal-parent-anchor" class="portal-parent" markdown="1">
<div class="portal-parent-fader-top"></div>
<div class="portal-parent-fader-bottom"></div><h1 id="cup"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∪</mo></mrow><annotation encoding="application/x-tex">\cup</annotation></semantics></math>∪<a aria-hidden="true" class="anchor-heading icon-link" href="#cup"></a></h1>
The union of two sets is the set consisting of elements from any of the sets.
<ul>
<li>in other words, the union set is composed of elements that exist in at least one of its constituent sets.</li>
<li>ex. if <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>5</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A = \{1, 3, 5, 7\}</annotation></semantics></math>A={1,3,5,7} and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">B = \{1, 2, 4, 6, 7\}</annotation></semantics></math>B={1,2,4,6,7}, then <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>4</mn><mo separator="true">,</mo><mn>5</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A \cup B = \{1, 2, 3, 4, 5, 6, 7\}</annotation></semantics></math>A∪B={1,2,3,4,5,6,7}</li>
</ul>
<h3 id="example-yelp">Example: Yelp<a aria-hidden="true" class="anchor-heading icon-link" href="#example-yelp"></a></h3>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">A =</annotation></semantics></math>A= The set of Yelp users who like Biz1</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">B =</annotation></semantics></math>B= The set of Yelp users who like Biz2</li>
</ul>
The set <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cup B</annotation></semantics></math>A∪B is the set of users that liked either Biz1 or Biz2 (or both)
</div></div>
<h2 id="probability-of-union-sets">Probability of Union sets<a aria-hidden="true" class="anchor-heading icon-link" href="#probability-of-union-sets"></a></h2>
The general probability addition rule for the union of two events states that 
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cup B) = P(A) + P(B) − P(A \cap B)</annotation></semantics></math>P(A∪B)=P(A)+P(B)−P(A∩B), where <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cap B</annotation></semantics></math>A∩B is the intersection of the two sets.</li>
</ul>
The addition rule can be shortened if the sets are disjoint (ie. have no members in common): 
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A \cup B) = P (A) + P (B)</annotation></semantics></math>P(A∪B)=P(A)+P(B)</li>
</ul>
can even be extended to more sets if they are all disjoint: 
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo>∪</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P( A \cup B \cup C) = P(A) + P(B) + P(C)</annotation></semantics></math>P(A∪B∪C)=P(A)+P(B)+P(C)</li>
</ul>

Union

thoughts-on


Welcome to my Second Brain! Here you'll find thoughts, lessons, facts and any other interesting musings that I've come across or thought up over the years. A lot of what I have gathered is originally from the minds of others, while some of it originates from myself. Some notes may contain misinformation, though I try to be rigorous with my additions.

What I have here does not necessarily reflect my thoughts and views. I collect information not based upon whether I agree with it or not, but rather when I determine it to contain some value that can be served in the future.  

You can think of this collection of inter-linked notes as my personal wiki.

This Dendron notebook is the sister vault to the tech-focused [Digital Garden](https://tech.kyletycholiz.com)

## Tags
Throughout the Second Brain, I have made use of tags, which give semantic meaning to the pieces of information.

- `ex.` - Denotes an *example* of the preceding piece of information
- `spec:` - Specifies that the preceding information has some degree of *speculation* to it, and may not be 100% factual. Ideally this gets clarified over time as my understanding develops. I try to go back after I have better understood the topic and clear out the notes of `spec:` tags
- `anal:` - Denotes an *analogy* of the preceding information. When I can, I attempt to link concepts to others that I have previously learned.
- `mn:` - Denotes a *mnemonic* to aid in memorization
- `expl:` - Denotes an *explanation*

## Resources
### UE (Unexamined) Resources
Often, I come across sources of information that I believe to be high-quality. They may be recommendations or found in some other way. No matter their origin, I may be in a position where I don't have the time to fully examine them (and properly extract notes), or I may not require the information at that moment in time. In cases like these, I will add reference to a section of the note called **UE Resources**. The idea is that in the future when I am ready to examine them, I have a list of resources that I can start with. This is an alternative approach to compiling browser bookmarks, which I've found can quickly become untenable.

### E (Examined) Resources
Once a resource has been thoroughly examined and has been mined for notes, it will be moved from *UE Resources* to *E Resources*. This is to indicate that (in my own estimation), there is nothing more to be gained from the resource that is not already in the note.

### Resources
This heading is for inexhaustible resources. 
- A prime example would be a quality website that continually posts articles- another example would be a tool, such as software that measures frequencies in a room to help acoustically treat it.


Union

∪\cup∪

Example: Yelp

Probability of Union sets

$\cup$