From 017d57987d2885fe8ad9038fb185de2822a71c8b Mon Sep 17 00:00:00 2001
From: benjitusk <benjitusk1@gmail.com>
Date: Wed, 24 Jan 2024 13:14:12 +0200
Subject: [PATCH] vault backup: 2024-01-24 13:14:12

---
 docs/Torah and Tech Week Four.pdf             |  Bin 0 -> 93356 bytes
 docs/Torah and Tech Week Three.pdf            |  Bin 0 -> 132744 bytes
 .../Homework/HW II - A Hash of the Clones.md  |  108 +
 .../Discrete Mathematics/01. Sets.md          |   10 +-
 .../Discrete Mathematics/02. Relations.md     |  139 +-
 .../Torah and Tech Week Four.md               |   28 +
 .../Torah and Tech Week Three.md              |   91 +
 ...elations 2024-01-23 17.38.17.excalidraw.md |  514 +++++
 ...elations 2024-01-23 17.46.05.excalidraw.md |  345 +++
 ...elations 2024-01-23 20.31.11.excalidraw.md | 1947 +++++++++++++++++
 .../Drawing 2024-01-22 11.18.38.excalidraw.md |  331 +++
 11 files changed, 3489 insertions(+), 24 deletions(-)
 create mode 100644 docs/Torah and Tech Week Four.pdf
 create mode 100644 docs/Torah and Tech Week Three.pdf
 create mode 100644 docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Four.md
 create mode 100644 docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Three.md
 create mode 100644 docs/excalidraw/02. Relations 2024-01-23 17.38.17.excalidraw.md
 create mode 100644 docs/excalidraw/02. Relations 2024-01-23 17.46.05.excalidraw.md
 create mode 100644 docs/excalidraw/02. Relations 2024-01-23 20.31.11.excalidraw.md
 create mode 100644 docs/excalidraw/Drawing 2024-01-22 11.18.38.excalidraw.md

diff --git a/docs/Torah and Tech Week Four.pdf b/docs/Torah and Tech Week Four.pdf
new file mode 100644
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diff --git a/docs/Year II/Semester I/Data Structures II/Homework/HW II - A Hash of the Clones.md b/docs/Year II/Semester I/Data Structures II/Homework/HW II - A Hash of the Clones.md
index 6b7644f..404663e 100644
--- a/docs/Year II/Semester I/Data Structures II/Homework/HW II - A Hash of the Clones.md	
+++ b/docs/Year II/Semester I/Data Structures II/Homework/HW II - A Hash of the Clones.md	
@@ -118,3 +118,111 @@ Since AVL-Search is $O(\log n)$, the worst case is $O(\log n)$.
 The worst case is where $|\text{tree}_{h(k)}|=n$.
 We need to delete from the AVL tree of size $n$ (height $\Theta(\log n)$).
 Since AVL-Delete is $O(\log n)$, the worst case is $O(\log n)$.
+
+
+### Question 4
+#### Part A
+```python
+def search(HT,k):
+	index=hash(k)
+	curr=HT[i]
+	while(concat(curr.l.bits, i.bits, curr.r.bits) != k*k):
+		curr = cur.next
+	return curr.val
+```
+
+The solution is to calculate the square of the key, instead of taking the square root of the concatenated bits. While the concatenated bits do not equal the square of the key, we have not yet found the value.
+
+#### Part B
+The size of the index is $p$ bits, because the hash table has $2^p$ entries.
+$b$ is the size of the key, therefore, $k^2$is $2b$ bits long.
+The following calculations are referring to the size (in bits), not the literal value.
+$2b=l+p+r$
+$l+r=2b-p$
+$2r=2b-p$ (because $l=r$)
+We save space when $2r<b$
+$2b-p<b$
+${b<p}$
+$\therefore$ We save memory space when $b<p$.
+
+### Question 5
+
+#### Linear Probing
+| Index | Content | Probes |
+| ---- | ---- | ---- |
+| 0 | 15 | 2 |
+| 1 | 1 | 1 |
+| 2 | 18 | 1 |
+| 3 | 34 | 2 |
+| 4 | 68 | 1 |
+| 5 | 51 | 3 |
+| 6 |  |  |
+| 7 | 39 | 1 |
+| 8 | 72 | 1 |
+| 9 |  |  |
+| 10 | 122 | 1 |
+| 11 | 43 | 1 |
+| 12 | 26 | 3 |
+| 13 | 13 | 1 |
+| 14 | 14 | 1 |
+| 15 | 31 | 1 |
+
+Average number of probes is $\frac{20}{14}=1.428$
+
+The expected runtime = $\Theta(1+\frac\alpha 2)$
+In our case, this is $1+\frac{14}{16\cdot2}=\frac{23}{16}=1.4375$
+The ratio is $\frac{20}{14}\div\frac{23}{16}=\frac{160}{161}=0.994$
+
+#### Quadratic Probing
+| Index | Content | Probes |
+| ---- | ---- | ---- |
+| 0 | 15 | 2 |
+| 1 | 1 | 1 |
+| 2 | 18 | 1 |
+| 3 | 34 | 2 |
+| 4 | 68 | 1 |
+| 5 |  |  |
+| 6 | 51 | 3 |
+| 7 | 39 | 1 |
+| 8 | 72 | 1 |
+| 9 | 26 | 6 |
+| 10 | 122 | 1 |
+| 11 | 43 | 1 |
+| 12 |  |  |
+| 13 | 13 | 1 |
+| 14 | 14 | 1 |
+| 15 | 31 | 1 |
+
+Average number of probes is $\frac{23}{14}=1.643$
+
+The expected runtime = $\Theta(1+\frac\alpha 2)$
+In our case, this is $1+\frac{14}{16\cdot2}=\frac{23}{16}=1.4375$
+The ratio is $\frac{23}{14}\div\frac{23}{16}=\frac{16}{14}=1.143$
+
+#### Double Hashing
+| Index | Content | Probes |
+| ---- | ---- | ---- |
+| 0 | 15 | 2 |
+| 1 | 1 | 1 |
+| 2 | 18 | 1 |
+| 3 | 51 | 1 |
+| 4 | 68 | 1 |
+| 5 |  |  |
+| 6 | 26 | 1 |
+| 7 | 34 | 2 |
+| 8 | 72 | 1 |
+| 9 | 43 | 2 |
+| 10 | 122 | 1 |
+| 11 | 39 | 3 |
+| 12 |  |  |
+| 13 | 13 | 1 |
+| 14 | 14 | 1 |
+| 15 | 31 | 1 |
+
+Average number of probes is $\frac{19}{14}=1.357$
+
+The expected runtime = $\Theta(1+\frac\alpha 2)$
+In our case, this is $1+\frac{14}{16\cdot2}=\frac{23}{16}=1.4375$
+The ratio is $\frac{19}{14}\div\frac{23}{16}=\frac{152}{161}=0.944$
+
+
diff --git a/docs/Year II/Semester I/Discrete Mathematics/01. Sets.md b/docs/Year II/Semester I/Discrete Mathematics/01. Sets.md
index f800394..acc0623 100644
--- a/docs/Year II/Semester I/Discrete Mathematics/01. Sets.md	
+++ b/docs/Year II/Semester I/Discrete Mathematics/01. Sets.md	
@@ -15,7 +15,7 @@
 - The **empty set** is the set containing no elements.
 	- Denoted as $\set{}$ or $\phi$ (Greek letter phi)
 	- It is a subset of *every* set, other than itself
-
+#Discrete-vocab
 
 > [!example] Set syntax and examples
 >
@@ -95,7 +95,7 @@ We must show that both of the following are true:
 *Dec. 31*
 
 ### Power Set
-**Definition:** The **power set** of a set $S$ (denoted $\pset S$) is the set of all subsets of $S$.
+**Definition:** The **power set** of a set $S$ (denoted $\pset S$) is the set of all subsets of $S$. #Discrete-vocab
 
 > [!example] Power Set Examples
 >
@@ -113,7 +113,7 @@ For any set $S$, $\phi\in\pset S$ and $S\in\pset S$.
 > $\set{1}\in\pset{\set{1,2}}$, but $1\notin\pset{\set{1,2}}$
 
 #### Power Set Theorem
-For any finite set $S$, $|\pset S|=2^{|S|}$, where $|S|$ is the number of elements in $S$.
+For any finite set $S$, $|\pset S|=2^{|S|}$, where $|S|$ is the number of elements in $S$. #Discrete-theorems
 
 This means that the number of elements in the power set of a set is equal to $2$ raised to the number of elements in the set.
 
@@ -134,7 +134,7 @@ This means that the number of elements in the power set of a set is equal to $2$
 		- Explanation: The intersection of all intervals $[\frac1i,\frac1{i+1})$ for $i=1,2,3,\dots$ is the set $\set{0}$ because the only number that is in every interval is $0$.
 
 ### Cartesian Products
-**Definition:** An **ordered pair** is a pair of elements $(a,b)$ in which the order of the elements matters. $(a,b)\not=(b,a)$.
+**Definition:** An **ordered pair** is a pair of elements $(a,b)$ in which the order of the elements matters. $(a,b)\not=(b,a)$. #Discrete-vocab
 
 **Definition:** An **ordered n-tuple** is an ordered list of $n$ elements $(a_1,a_2,\dots,a_n)$ in which the order of the elements matters. $(a_1,a_2,\dots,a_n)\not=(a_2,a_1,\dots,a_n)$.
 
@@ -143,7 +143,7 @@ Familiar examples of ordered pairs and ordered n-tuples:
 - Point in space: $(x,y,z)$
 
 **Definition:**
-Let $A, B$ be sets. The **Cartesian product** of $A$ and $B$ (denoted $A\times B$) is the set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.
+Let $A, B$ be sets. The **Cartesian product** of $A$ and $B$ (denoted $A\times B$) is the set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$. #Discrete-vocab
 
 Formally, $A\times B=\set{(a,b)\mid a\in A,\ b\in B}$
 
diff --git a/docs/Year II/Semester I/Discrete Mathematics/02. Relations.md b/docs/Year II/Semester I/Discrete Mathematics/02. Relations.md
index 15a12ec..f95ae78 100644
--- a/docs/Year II/Semester I/Discrete Mathematics/02. Relations.md	
+++ b/docs/Year II/Semester I/Discrete Mathematics/02. Relations.md	
@@ -1,6 +1,5 @@
-*Dec. 31*
+*Jan 7*
 ## Relations
-
 Intuitively, a relation is a way of "comparing" two elements
 1. $<$, $>$, $=$, $\leq$, $\geq$, among numbers
 2. $A=$ set of people, $B=$ set of cities
@@ -8,7 +7,8 @@ Intuitively, a relation is a way of "comparing" two elements
 3. $A=$ set of people
   - We can define a relation to hold between person $a_1\in A$ and person $a_2\in A$ if they share a family connection.
 
-Formal **Definition:** Let $A,B$ be sets. A **relation** $R$ from $A$ to $B$ is a subset of $A\times B$.
+**Formal Definition:**
+Let $A,B$ be sets. A **relation** $R$ from $A$ to $B$ is a subset of $A\times B$.
 
 _Yeah, that's it. The relation doesn't have to be anything meaningful. It's just a subset of the Cartesian product of the two sets._
 
@@ -20,7 +20,7 @@ _Yeah, that's it. The relation doesn't have to be anything meaningful. It's just
 > 2. "greater than" relation: $R=\set{(2,1), (3,1), (3,2)}$
 > 3. "arbitrary" relation: $R=\set{(1,1), (1,3), (2,1)}$
 
-A relation $R$ from $A$ to itself (i.e. $R\sseq A\times A$) is called a **relation on $A$**.
+A relation $R$ from $A$ to itself (i.e. $R\sseq A\times A$) is called a **relation on $A$**. #Discrete-vocab
 
 For example, see the third example in the intuitive definition above.
 
@@ -34,10 +34,11 @@ For example, see the third example in the intuitive definition above.
   - E.g, $A=\set{1,2,3}$, $B=\set{4,5,6}$, $R=$ "less than"
   - $R=A\times B$ because every element of $A$ is less than every element of $B$
 
-The ususal set operations can be applied to relations, e.g. union, intersection, complement, etc. since relations are just sets.
+The usual set operations can be applied to relations, e.g. union, intersection, complement, etc. since relations are just sets.
 
 ### Inverse Relations
-**Definition:** Let $R$ be a relation from $A$ to $B$. The **inverse relation** of $R$ (denoted $R^{-1}$) is the relation from $B$ to $A$ defined by $R^{-1}=\set{(b,a)\mid(a,b)\in R}$.
+**Definition:**
+Let $R$ be a relation from $A$ to $B$. The **inverse relation** of $R$ (denoted $R^{-1}$) is the relation from $B$ to $A$ defined by $R^{-1}=\set{(b,a)\mid(a,b)\in R}$. #Discrete-vocab
 
 > [!example] Examples of Inverse Relations
 >
@@ -50,11 +51,12 @@ The ususal set operations can be applied to relations, e.g. union, intersection,
 ### Composition of Relations
 Suppose $R_1 \sseq A\times B$ and $R_2 \sseq B\times C$ are relations.
 
-**Definition:** The **composition** of $R_1$ and $R_2$ (denoted $R_1\circ R_2$) is the relation from $A$ to $C$ defined by $R_1\circ R_2=\set{(a,c)\mid\exists b\in B\text{ s.t. }(a,b)\in R_1\text{ and }(b,c)\in R_2}$.
+**Definition:**
+The **composition** of $R_1$ and $R_2$ (denoted $R_1\circ R_2$) is the relation from $A$ to $C$ defined by $R_1\circ R_2=\set{(a,c)\mid\exists b\in B\text{ s.t. }(a,b)\in R_1\text{ and }(b,c)\in R_2}$.
 
 > [!tldr] Compositions explained in English
 >
-> A composition of relations $R_1$ and $R_2$ is a new relation $R_1\circ R_2$ that contains all pairs $(a,c)$ where there exists some $b$ such that $(a,\boldsymbol{b})\in R_1$ and $(\boldsymbol{b},c)\in R_2$.
+> A composition of relations $R_1$ and $R_2$ is a new relation $R_1\circ R_2$ that contains all pairs $(a,c)$ where there exists some $b$ such that $(a,\boldsymbol{b})\in R_1$ and $(\boldsymbol{b},c)\in R_2$. #Discrete-vocab
 
 > [!example] Examples of common composed relations
 >
@@ -70,15 +72,107 @@ Suppose $R_1 \sseq A\times B$ and $R_2 \sseq B\times C$ are relations.
 - **Associative:** $(R_1\circ R_2)\circ R_3=R_1\circ(R_2\circ R_3)$
 
 ### Equivalence Relations
-> [!error] Missing section
+#### Partitions
+**Definition:**
+Let $A$ be any set. A **partition** $P$ of $A$ is a set of subsets of $A$: 
+
+$$P=\set{A_1,A_2,\dots,A_i,\dots,A_n}$$
+such that two conditions are met: #Discrete-vocab
+1. For all $i\not=j:A_i\cap A_j=\phi$
+	- Subsets in $P$ are disjoint
+2. $\cup_i^nA_i=A$
+	- Every element of $A$ is in some subset of the partition
+
+>[!example] Example of a partition
+>
+> Let $A=\set{1,2,3,4,5,6,7}$
+> Let $A_1=\set{2,5}$
+> Let $A_2=\set{1,3,6}$
+> Let $A_3=\set{7}$
+> Let $A_4=\set{4}$
+>
+> Then the partition $P$ of $A$ looks like this:
+> 
+> ![[Excalidraw/02. Relations 2024-01-23 17.38.17.excalidraw]]
+
+#### Equivalence Relations
+**Definition** Given a partition $P$ of $A$, define a relation $R$ as follows:
+$(a,b)\in R$ whenever $a,b$ belong to the same subset in the partition $P$.
+#Discrete-vocab 
+
+In our *above* example, $(1,6)\in R$, $(1,2)\in R$, $(7,7)\in R$, $(7,6)\not\in R$.
+
+> [!example] Example of an Equivalence Relation
+> 
+> Let $A=\set{1,2,3,4}$
+> Let the partition be defined as specified in this diagram:
+> ![[Excalidraw/02. Relations 2024-01-23 17.46.05.excalidraw]]
+> 
+> Then, $R=\set{(1,3),(3,1),(2,2),(4,4),(1,1),(3,3)}$
+
+$R$ is called an *equivalence relation*, and we say that $R$ is *induced* by the partition $P$. #Discrete-vocab
+
+Any relation $R$ on $A$ is:
+- Reflexive
+	- Every $a\in A$ belongs to the same “cell” as itself
+- Symmetric
+	- If $a$ is in $b$’s cell, then $b$ is in $a$’s cell
+- Transitive
+	- If $a$ is in $b$’s cell, and $b$ is in $c$’s cell, then $a$ is in $c$’s cell
+#### Equivalence Relation Theorem
+Any relation $R$ on $A$ which is reflexive, symmetric, and transitive is an equivalence relation (induced by some partition of $A$).
+
+I.e., we can “work backwards” from a relation satisfying these three properties to find the partition which induced it.
+#Discrete-theorems 
+
+> [!example] Example of the Equivalence Relation Theorem
+> 
+> $A=\Z$
+> $R\subseteq \Z\times\Z$
+> $R=\set{(a,b)\mid a-b \mod 5=0}$
+> $(3,-2)\in R, (5,0)\in R, (-7,3)\in R, (3,13)\in R$
+> $(1,2)\not\in R, (4,0)\not\in R, (2,-7)\not\in R, (8,8)\in R$
+> 
+> **Claim:** $R$ is reflexive, symmetric, and transitive.
+> 
+> **Proof:**
+> 1. $\forall n\in\Z,n-n=0=0\cdot5$; hence $(n,n)\in R$
+> 2. If $a-b=5k$ then $b-a=-5k=5\cdot(-k)$
+> 	- If $(a,b)\in R$ then $(b,a)\in R$
+> 3. If $a-b=5k$ and $b-c=5l$, then $a-c=(a-b)+(b-c)=5k+5l=5(k+l)\Rightarrow(a,c)\in R$
+>  
+> Therefore, $R$ is an equivalence relation.   
+
+#### Equivalence Classes
+**Definition:**
+The cells of a partition (which induces $R$) are called the **equivalence classes** of $R$.
+
+For each $a\in A$, $[a]$ means the equivalence class of $a$.
+E.g., in the last example, $[3]=[-12]$ (because they are both two less than multiples of 5)
+
+The set of all equivalence classes of $A$ in the relation $R$ is denoted $A/R$ (“$A$ modulo $R$”) and is called the *quotient set* of $R$. #Discrete-vocab 
+
+In the last example, $A/R=\set{[0],[1],[2],[3],[4],[5]}$
+
+> [!example] Example of Quotient Sets
+> $A=$ set of people
+> $R$ is a relation on $A$: $(a,b)\in R$ when $a,b$ share last names.
+> 
+> It’s easy to see that $R$ is reflexive, symmetric, and transitive, and hence is an equivalence relation.
+>
+>It’s also easy to see that each equivalence class corresponds to a family name.
+>
+>The quotient set $A/R$ is the set of all last names.
+
+
+*Jan 14*
 ### Order Relations
+**Definition:**
+An order relation generalizes the familiar “orders” like ≤ or ≥ on sets of numbers. #Discrete-vocab Formally, a relation $R$ on a set $A$ is an **order relation** if it has three properties:
 #### Properties of Order Relations
-An order relation generalizes the familiar “orders” like ≤ or ≥ on sets of numbers.
-Formally, a relation $R$ on a set $A$ is an **order relation** if it has three properties:
-
 1. $R$ is reflexive
 2. $R$ is transitive
-3. $R$ is antisymmetric
+3. $R$ is antisymmetric #Discrete-vocab 
 	- i.e., if $(a,b)\in R$ and $(b,a)\in R$, then $a=b$
 	- $R$ cannot hold between different elements in both directions
 	- Note that this is **not** the same as being asymmetric
@@ -87,6 +181,7 @@ Formally, a relation $R$ on a set $A$ is an **order relation** if it has three p
 If $R$ is an order relation, then we use the notation
 
 $$a\leqslant_Rb$$
+in place of using $aRb$ or $(a,b)\in R$.
 
 > [!example] Examples of order relations
 > 
@@ -98,21 +193,27 @@ $$a\leqslant_Rb$$
 >  4.  Let $A=\N$.
 >     Define $R$ on $A$ as: $(a,b)\in R$ if $a|b$ ($a$ divides $b$).
 
-Note that examples 1 and 2 are “stronger” orderings. They are called **total orders** because $\forall a \in A$, $\exists b$ s.t. either $(a,b)\in R$ or $(b,a)\in R$.
-Another way to say this is $\forall a \in A,\exists b$ s.t. $(a,b)\in R\cup R^{-1}$
-Examples 3 and 4 are **partial orders** because there exist at least two elements in $A$ that cannot be compared with each other.
+**Definition:**
+An order relation $R$ on a set $A$ is a **total order** if, for *every* $a_1,a_2\in A$, either $a_1\leq_Ra_2$ or $a_2\leq_Ra_1$ (or both, if $a=b$).
+Otherwise, $R$ is called a **partial order**. #Discrete-vocab
+
+Examples 1 and 2 above are examples of total orders, and examples 3 and 4 above are partial orders.
 
 ##### Hasse Diagrams
-A partial order $R$ on a set $A$ can be illustrated using a **Hasse diagram**:
+A partial order $R$ on a set $A$ can be illustrated using a **Hasse diagram**.
 
 For example, $A={n|1≤n≤20}$, and $R$ is the divisibility relation (example 3 from above)
 
-###### Property of HD
+![[Excalidraw/02. Relations 2024-01-23 20.31.11.excalidraw]]
+
+###### Property of Hasse Diagrams
 We can get from $a_1$ to $a_2$ by moving up edges iff then $a_1\leq_Ra_2$
 ###### Rules of HD
 1. All **minimal elements** go on the bottom.
 	- ($a_1 \in A$ is minimal if there does $\not\exists b\in A$ s.t. $b\leq_Ra$)
-	- If there is a single minimum element $a$, then $a$ is called a minimum in $R$.
+	- If there is a single minimum element $a$, then $a$ is called a minimum in $R$. #Discrete-vocab 
+	- In the above example, $1$ is a minimum.
+		- If we delete $1$ from $A$, then $2,3,5,7,11,13,17,19$ are all minimal elements, but there is no *minimum*.
 2. An edge exists from $a_1$ “goes up” to $a_2$ if:
 	1. $a_1 \not= a_2$
 	2. $a_1\leq_Ra_2$
diff --git a/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Four.md b/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Four.md
new file mode 100644
index 0000000..c87f22e
--- /dev/null
+++ b/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Four.md	
@@ -0,0 +1,28 @@
+*By Benji Tusk*
+
+**Prohibited Sources (i) – The Status of the Stem Cells**
+
+1. **Mere Water (מיא בעלמא):**
+   - Rabbi Zvi Ryzman, in his essay, argues that stem cells used in cultured meat should be considered as "mere water" and not actual meat.
+   - He bases this argument on the Talmudic concept that within the first forty days of gestation, a fetus is not considered a viable human life, akin to "mere water."
+   - The rationale is that until forty days, the fetus lacks sufficient development to be considered meat, and therefore, stem cells before this period might be viewed similarly.
+
+2. **Objections to the 'Mere Water' Concept:**
+   - Rabbi Yaakov Ariel objects to this perspective. He argues that the growth of microscopic entities doesn't exempt them from being considered meat. While they may not be fit for consumption due to their size, their essence remains meat-like.
+   - Another objection suggests that the 'mere water' concept is only applicable when the forbidden substance is not present within the growing entity. Stem cells, on the other hand, maintain the imprint of their origin, potentially containing DNA from a non-kosher source.
+
+3. **Comparison to Agricultural Produce:**
+   - Some suggest a comparison to "outgrowths of teruma," where the halachic status of a seed does not necessarily transfer to its produce. This principle is found in Mishna Terumot.
+   - The Mishna distinguishes between seeds that rot and disintegrate, nullifying the prohibition, and corms where the core remains for the next season, making their outgrowths prohibited.
+
+4. **Rambam's Perspective on Growth:**
+   - Rambam's view is introduced, suggesting that the growth process changes the halachic status of the plant, even if the root does not disintegrate. This is relevant to the laws of teruma and untithed produce.
+   - Rabbi Ariel, however, contends that stem cells maintain the forbidden substance within their structure, potentially through the DNA code.
+
+5. **The 'Mere Water' Concept and Abortion:**
+   - The concept of "mere water" is brought into the discussion of abortion. Some poskim debate whether a human fetus is considered 'mere water' up to forty days from conception, potentially allowing abortion up to that point or under special circumstances.
+   - Various poskim, such as Rabbi Yair Chayim Bacharach and Rabbi Moshe Feinstein, present differing views on the permissibility of abortion based on this concept.
+
+**Conclusion:**
+   - The application of the 'mere water' concept to stem cells in the context of cultured meat is met with various objections. The presence of the forbidden substance and the potential impact of growth on the halachic status of the cells are central issues in this debate.
+   - The comparison to agricultural produce, Rambam's perspective on growth, and the application of 'mere water' to abortion further complicate the discussion, highlighting the intricate nature of applying Halacha to emerging technologies and ethical dilemmas.
\ No newline at end of file
diff --git a/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Three.md b/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Three.md
new file mode 100644
index 0000000..c06a09c
--- /dev/null
+++ b/docs/Year II/Semester I/Torah and Tech/Torah and Tech Week Three.md	
@@ -0,0 +1,91 @@
+Miraculously-Created Meat
+
+1. תענית כה ע"א:
+    - Rabbi Chanina ben Dosa observed a unique situation with his daughter experiencing distress.
+    - He inquired about the cause, and she explained that her vinegar jug had been swapped for one filled with oil, allowing her to kindle the Sabbat lights.
+    - Rabbi Chanina replied indifferently, suggesting that if oil can be used, then vinegar should also be permitted to ignite.
+    - The Talmud records a case where a miraculous flame burned continuously until they brought it for the Sabbat blessing.
+
+When cultured meat emerged, discussions arose on its halachic status, covering several aspects:
+
+1. Consumption of meat from non-kosher sources using this technology.
+2. Sourcing meat from living animals through biopsy instead of traditional slaughter.
+3. Concerns about the kashrut of nutrients used.
+4. Determining if cultured meat is parev or classified as actual meat.
+
+Rabbi Zvi Ryzman proposed a unique perspective, suggesting that the source of stem cells was irrelevant due to the extra-natural process, comparing it to miraculously created meat. Rabbi Chanoch Kahn's essay explored various approaches, focusing on the identity of cultured meat.
+
+1. Food Descended from Heaven
+    
+    - The Gemara discusses Adam's prohibition from eating meat until Noah's time when it was permitted.
+    - Rabbi Yehuda ben Tema suggests that angels used to roast meat for Adam in the Garden of Eden.
+    - Resolving the contradiction, the Gemara states that meat descending from heaven is not prohibited.
+    - Rashi and the Ran differ on the reasoning, with the latter suggesting it's not meat according to halacha.
+    - Chazon Ish proposes that heavenly involvement transforms the meat's status.
+2. Heavenly Wheat in Menachot
+    
+    - Rabbi Zira questions the status of wheat that descended with rain.
+    - Rashi explains it as wheat carried by rain, and Tosafot suggests it miraculously descended.
+    - The difference lies in whether it was a natural occurrence or a miraculous event.
+    - The discussion relates to whether "ממושבותיכם" in the Torah is satisfied by wheat descending in an abnormal manner.
+3. Miraculously Multiplied Oil
+    
+    - The Talmud recounts the miracle of Chanukah where a small amount of oil burned for eight days.
+    - Acharonim debate whether the oil was consumed gradually or replenished miraculously.
+    - In a similar narrative with Elisha, a widow's oil miraculously multiplied to pay her debts.
+    - Rashi links this story to the widow of Obadjahu, who hid prophets, suggesting a miraculous element.
+    - The discussion revolves around whether miraculously multiplied oil is fit for the Menorah but exempt from tithes.
+#### 1. Gemara Stories:
+
+The Gemara stories cited involve instances where animals were supposedly created through spiritual powers, akin to using the Sefer Yetzira. These animals were considered kosher, which raises the question of whether this could serve as a precedent for cultured meat. It's crucial to note that the Gemara shared these stories to argue against the prohibition of witchcraft in using the Sefer Yetzira.
+
+#### 2. Malbim's Question:
+
+The Malbim's perspective suggests that meat created through the Sefer Yetzira has no historical background and, therefore, might not be subject to certain prohibitions, such as the prohibition of consuming meat and milk together. The text asks for other perspectives on the Malbim's question.
+
+#### 3. Shelah's Interpretation:
+
+The Shelah suggests that Joseph's brothers were created through the Sefer Yetzira, possibly exempting them from the prohibition of eating meat from a living animal. However, objections are raised, and there's a query about the authenticity of such miraculous occurrences.
+
+#### 4. Objections by פתחי תשובה:
+
+The פתחי תשובה raises objections against the Shelah's opinion, particularly concerning the need for proper slaughter. These objections may stem from concerns related to the essence of reading the Torah and how it aligns with the opinion of the Ran mentioned in the first source.
+
+#### 5. Further Halachic Implications:
+
+Later authorities, like Rabbi Zvi Hirsch Ashkenazi (חכם צבי), express uncertainties regarding the halachic status of a person created through the Sefer Yetzira. The Chazon Ish and others raise concerns about the neveila status of such meat, asserting it may not be subject to the laws of ritual slaughter.
+
+### Responses to Questions:
+
+#### 1. The פתחי תשובה's Objections:
+
+The פתחי תשובה objects to the Shelah's opinion, primarily concerning the need for slaughter and raises questions about the essence of reading the Torah. These objections may align with or contradict the Ran's opinion from the first source, depending on their stance regarding miraculous occurrences.
+
+#### 2. Chazon Ish's Approach:
+
+The Chazon Ish offers a novel perspective suggesting that meat created through Kabbalistic means might be exempt from the prohibition of neveila but still considered meat. The question is whether this aligns with the Gemara he cites as a proof.
+
+3. **Is Cultured Meat Created Miraculously?**
+
+Rav Asher Weiss rejected the comparison of cultured meat to the cases discussed above.
+
+14. **Rav Asher Weiss's Objection:**
+   - Even if one accepts that meat created miraculously through a Sefer Yetzira-like process doesn't require ritual slaughter and is not subject to the prohibition of mixing meat and milk, Rav Asher Weiss disagrees with applying the same reasoning to cultured meat.
+   - He argues that cultured meat, despite its unique production method, is not a miraculous creation but rather an accelerated natural process.
+   - Cultured meat, according to Weiss, originates from natural flesh, growing and multiplying according to the laws of nature, without the intervention of supernatural forces.
+   - Hence, Weiss concludes that it doesn't fall under the category of miraculous creation and should be judged according to the general rules governing meat.
+
+15. **Rabbi Betzalel Stern's Distinction:**
+   - Rabbi Betzalel Stern introduces a crucial distinction between items created miraculously by humans and those created by God.
+   - He argues that when something is miraculously created by humans, like the meat that descended from heaven, it does not bear the same legal weight as naturally occurring phenomena or those created by God.
+   - However, creations that are truly miraculous and originate solely from divine intervention, without human involvement, are subject to all the commandments.
+   - Stern suggests that for the Almighty, who operates outside the confines of nature, there is no difference between miracles and the natural order; both are His creations.
+   - Therefore, when God provides man with food, whether miraculously or through nature, it is considered food by all standards.
+   
+**Reflection on Cultured Meat:**
+   - Applying these distinctions to cultured meat, Rav Asher Weiss views it as a product of accelerated natural processes rather than a miraculous creation.
+   - If one accepts Weiss's perspective, it aligns more closely with Stern's notion of creations with human involvement, deviating from the natural order but not qualifying as true miracles.
+   - Cultured meat, under this interpretation, might be considered a technological advancement, resembling ordinary daily miracles rather than a supernatural event.
+
+**Conclusion:**
+   - The debate on the status of cultured meat is based on whether it is a product of human activity or an accelerated natural process.
\ No newline at end of file
diff --git a/docs/excalidraw/02. Relations 2024-01-23 17.38.17.excalidraw.md b/docs/excalidraw/02. Relations 2024-01-23 17.38.17.excalidraw.md
new file mode 100644
index 0000000..11fc318
--- /dev/null
+++ b/docs/excalidraw/02. Relations 2024-01-23 17.38.17.excalidraw.md	
@@ -0,0 +1,514 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
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+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": 25.070583741336748,
+			"y": 51.219846792021684,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "#ffffff",
+			"width": 12.79998779296875,
+			"height": 25,
+			"seed": 894373059,
+			"groupIds": [
+				"tLMhiTgMgRBGEih4u5Z41"
+			],
+			"frameId": null,
+			"roundness": null,
+			"boundElements": [],
+			"updated": 1706024479726,
+			"link": null,
+			"locked": false,
+			"fontSize": 20,
+			"fontFamily": 1,
+			"text": "4",
+			"rawText": "4",
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"containerId": null,
+			"originalText": "4",
+			"lineHeight": 1.25,
+			"baseline": 18
+		},
+		{
+			"type": "text",
+			"version": 244,
+			"versionNonce": 520744941,
+			"isDeleted": false,
+			"id": "xHVK1kb2",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": 25.30603888538161,
+			"y": 104.80480310441084,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "#ffffff",
+			"width": 10.759994506835938,
+			"height": 25,
+			"seed": 1129885453,
+			"groupIds": [
+				"tLMhiTgMgRBGEih4u5Z41"
+			],
+			"frameId": null,
+			"roundness": null,
+			"boundElements": [],
+			"updated": 1706024482414,
+			"link": null,
+			"locked": false,
+			"fontSize": 20,
+			"fontFamily": 1,
+			"text": "7",
+			"rawText": "7",
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"containerId": null,
+			"originalText": "7",
+			"lineHeight": 1.25,
+			"baseline": 18
+		}
+	],
+	"appState": {
+		"theme": "dark",
+		"viewBackgroundColor": "#ffffff",
+		"currentItemStrokeColor": "#1e1e1e",
+		"currentItemBackgroundColor": "#ffffff",
+		"currentItemFillStyle": "solid",
+		"currentItemStrokeWidth": 2,
+		"currentItemStrokeStyle": "solid",
+		"currentItemRoughness": 1,
+		"currentItemOpacity": 100,
+		"currentItemFontFamily": 1,
+		"currentItemFontSize": 20,
+		"currentItemTextAlign": "left",
+		"currentItemStartArrowhead": null,
+		"currentItemEndArrowhead": "arrow",
+		"scrollX": 506.8657065962337,
+		"scrollY": 167.4834998938901,
+		"zoom": {
+			"value": 1.3248837049588957
+		},
+		"currentItemRoundness": "round",
+		"gridSize": null,
+		"gridColor": {
+			"Bold": "#C9C9C9FF",
+			"Regular": "#EDEDEDFF"
+		},
+		"currentStrokeOptions": null,
+		"previousGridSize": null,
+		"frameRendering": {
+			"enabled": true,
+			"clip": true,
+			"name": true,
+			"outline": true
+		}
+	},
+	"files": {}
+}
+```
+%%
\ No newline at end of file
diff --git a/docs/excalidraw/02. Relations 2024-01-23 17.46.05.excalidraw.md b/docs/excalidraw/02. Relations 2024-01-23 17.46.05.excalidraw.md
new file mode 100644
index 0000000..7fd10f2
--- /dev/null
+++ b/docs/excalidraw/02. Relations 2024-01-23 17.46.05.excalidraw.md	
@@ -0,0 +1,345 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
+
+
+# Text Elements
+4 ^LI6N3o36
+
+3 ^P8CYveTj
+
+2 ^4S9EyIw9
+
+1 ^VnKfbKO0
+
+%%
+# Drawing
+```json
+{
+	"type": "excalidraw",
+	"version": 2,
+	"source": "https://github.com/zsviczian/obsidian-excalidraw-plugin/releases/tag/2.0.17",
+	"elements": [
+		{
+			"id": "G_2nnR4Zi1czYHNmyIlX2",
+			"type": "rectangle",
+			"x": -170.325927734375,
+			"y": -18.9002685546875,
+			"width": 279.0001220703125,
+			"height": 129.86004638671875,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"RwZTppDe0YH618mXLhRhW"
+			],
+			"frameId": null,
+			"roundness": {
+				"type": 3
+			},
+			"seed": 2109489406,
+			"version": 94,
+			"versionNonce": 70712190,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024791206,
+			"link": null,
+			"locked": false
+		},
+		{
+			"id": "6WkIVLczQkAaMFTlgvDEv",
+			"type": "line",
+			"x": -23.7705078125,
+			"y": -16.552734375,
+			"width": 0.487060546875,
+			"height": 129.55731201171875,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"RwZTppDe0YH618mXLhRhW"
+			],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 1632902974,
+			"version": 97,
+			"versionNonce": 996606818,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024791207,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					0.487060546875,
+					129.55731201171875
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "JOhPziXKFTjmoNVB16TNa",
+			"type": "line",
+			"x": -24.14849853515625,
+			"y": 49.68035888671875,
+			"width": 132.9754638671875,
+			"height": 2.31646728515625,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"RwZTppDe0YH618mXLhRhW"
+			],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 1804087266,
+			"version": 41,
+			"versionNonce": 1426512830,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024791207,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					132.9754638671875,
+					-2.31646728515625
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "VnKfbKO0",
+			"type": "text",
+			"x": -115.26446533203125,
+			"y": 36.97943115234375,
+			"width": 5.4199981689453125,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"TmsRj1kXe6rzc3btn17a2"
+			],
+			"frameId": null,
+			"roundness": null,
+			"seed": 499170750,
+			"version": 27,
+			"versionNonce": 1570165438,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024825350,
+			"link": null,
+			"locked": false,
+			"text": "1",
+			"rawText": "1",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "1",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "P8CYveTj",
+			"type": "text",
+			"x": -74.4681396484375,
+			"y": 57.22576904296875,
+			"width": 13.6199951171875,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"iPLKY8jtj36kinWLjd5Tg"
+			],
+			"frameId": null,
+			"roundness": null,
+			"seed": 506150818,
+			"version": 41,
+			"versionNonce": 1515334846,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024823670,
+			"link": null,
+			"locked": false,
+			"text": "3",
+			"rawText": "3",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "3",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "4S9EyIw9",
+			"type": "text",
+			"x": 37.20184326171875,
+			"y": 0.8961181640625,
+			"width": 14.239990234375,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"CbWd4JFukr7ShqB4f-V-H"
+			],
+			"frameId": null,
+			"roundness": null,
+			"seed": 640560510,
+			"version": 63,
+			"versionNonce": 339470690,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024809563,
+			"link": null,
+			"locked": false,
+			"text": "2",
+			"rawText": "2",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "2",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "LI6N3o36",
+			"type": "text",
+			"x": 35.48248291015625,
+			"y": 63.15008544921875,
+			"width": 12.79998779296875,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [
+				"RwZTppDe0YH618mXLhRhW"
+			],
+			"frameId": null,
+			"roundness": null,
+			"seed": 767640162,
+			"version": 87,
+			"versionNonce": 537059390,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706024817952,
+			"link": null,
+			"locked": false,
+			"text": "4",
+			"rawText": "4",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "4",
+			"lineHeight": 1.25
+		}
+	],
+	"appState": {
+		"theme": "dark",
+		"viewBackgroundColor": "#ffffff",
+		"currentItemStrokeColor": "#1e1e1e",
+		"currentItemBackgroundColor": "transparent",
+		"currentItemFillStyle": "solid",
+		"currentItemStrokeWidth": 2,
+		"currentItemStrokeStyle": "solid",
+		"currentItemRoughness": 1,
+		"currentItemOpacity": 100,
+		"currentItemFontFamily": 1,
+		"currentItemFontSize": 20,
+		"currentItemTextAlign": "left",
+		"currentItemStartArrowhead": null,
+		"currentItemEndArrowhead": "arrow",
+		"scrollX": 511,
+		"scrollY": 360.9765625,
+		"zoom": {
+			"value": 1
+		},
+		"currentItemRoundness": "round",
+		"gridSize": null,
+		"gridColor": {
+			"Bold": "#C9C9C9FF",
+			"Regular": "#EDEDEDFF"
+		},
+		"currentStrokeOptions": null,
+		"previousGridSize": null,
+		"frameRendering": {
+			"enabled": true,
+			"clip": true,
+			"name": true,
+			"outline": true
+		}
+	},
+	"files": {}
+}
+```
+%%
\ No newline at end of file
diff --git a/docs/excalidraw/02. Relations 2024-01-23 20.31.11.excalidraw.md b/docs/excalidraw/02. Relations 2024-01-23 20.31.11.excalidraw.md
new file mode 100644
index 0000000..34c05fb
--- /dev/null
+++ b/docs/excalidraw/02. Relations 2024-01-23 20.31.11.excalidraw.md	
@@ -0,0 +1,1947 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
+
+
+# Text Elements
+1 ^uwUSFDbX
+
+2 ^DGr5PfKJ
+
+3 ^ut4PL0AE
+
+4 ^2XfA0DLR
+
+5 ^pCBL91l5
+
+6 ^vElSdqCp
+
+7 ^H3ut7blq
+
+8 ^2FUuzJky
+
+9 ^M7C0yq5r
+
+10 ^aIvn1hef
+
+11 ^ALGc06xp
+
+12 ^AT2Ji1fF
+
+13 ^s1czTQ5B
+
+14 ^7feWFdAp
+
+15 ^347xNLpy
+
+16 ^KCbmuMZz
+
+17 ^A4aZ2eMD
+
+18 ^sTz8nuDg
+
+19 ^SOrD46qt
+
+20 ^46Yfkoc8
+
+%%
+# Drawing
+```json
+{
+	"type": "excalidraw",
+	"version": 2,
+	"source": "https://github.com/zsviczian/obsidian-excalidraw-plugin/releases/tag/2.0.17",
+	"elements": [
+		{
+			"id": "uwUSFDbX",
+			"type": "text",
+			"x": -192.51043701171875,
+			"y": 132.08798217773438,
+			"width": 5.4199981689453125,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 1461405591,
+			"version": 139,
+			"versionNonce": 362439033,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706034971602,
+			"link": null,
+			"locked": false,
+			"text": "1",
+			"rawText": "1",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "1",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "DGr5PfKJ",
+			"type": "text",
+			"x": -390.4864614386307,
+			"y": 84.46664428710938,
+			"width": 14.239990234375,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 105305081,
+			"version": 150,
+			"versionNonce": 1487859929,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706034935352,
+			"link": null,
+			"locked": false,
+			"text": "2",
+			"rawText": "2",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "2",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "ut4PL0AE",
+			"type": "text",
+			"x": -335.30789551699064,
+			"y": 84.46664428710938,
+			"width": 13.6199951171875,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 2073744823,
+			"version": 132,
+			"versionNonce": 1213100727,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706034942965,
+			"link": null,
+			"locked": false,
+			"text": "3",
+			"rawText": "3",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "3",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "2XfA0DLR",
+			"type": "text",
+			"x": -417.9078706440173,
+			"y": 29.753997802734375,
+			"width": 12.79998779296875,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 19842009,
+			"version": 181,
+			"versionNonce": 588305593,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706034952580,
+			"link": null,
+			"locked": false,
+			"text": "4",
+			"rawText": "4",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "4",
+			"lineHeight": 1.25
+		},
+		{
+			"id": "pCBL91l5",
+			"type": "text",
+			"x": -280.74932471253805,
+			"y": 84.46664428710938,
+			"width": 12.3599853515625,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 323860887,
+			"version": 161,
+			"versionNonce": 1079353,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706034942965,
+			"link": null,
+			"locked": false,
+			"text": "5",
+			"rawText": "5",
+			"fontSize": 20,
+			"fontFamily": 1,
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+				"type": 2
+			},
+			"seed": 1981752663,
+			"version": 35,
+			"versionNonce": 801325177,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035070876,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					13.475616455078125,
+					22.24127197265625
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "6Hrzb5N55wSgCp7B2F5Di",
+			"type": "line",
+			"x": -336.9625549316406,
+			"y": 0.0283203125,
+			"width": 17.094635009765625,
+			"height": 27.98382568359375,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 0.5,
+			"strokeStyle": "solid",
+			"roughness": 0,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 2052507993,
+			"version": 30,
+			"versionNonce": 1718848567,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035072930,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					-17.094635009765625,
+					27.98382568359375
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "NE1L2Ke5Bnay58Ib_D8NL",
+			"type": "line",
+			"x": -330.5239562988281,
+			"y": 3.6971435546875,
+			"width": 15.869293212890625,
+			"height": 25.6751708984375,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 0.5,
+			"strokeStyle": "solid",
+			"roughness": 0,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 1052962135,
+			"version": 34,
+			"versionNonce": 934597529,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035075487,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					15.869293212890625,
+					25.6751708984375
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "FLhCkGReDEBmGm30l90eF",
+			"type": "line",
+			"x": -456.71002197265625,
+			"y": -43.0654296875,
+			"width": 11.665496826171875,
+			"height": 21.57525634765625,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 0.5,
+			"strokeStyle": "solid",
+			"roughness": 0,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 820574329,
+			"version": 42,
+			"versionNonce": 916013369,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035082431,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					11.665496826171875,
+					21.57525634765625
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "Giqe2yEFjyh6gzhgW7K0H",
+			"type": "line",
+			"x": -399.6334228515625,
+			"y": 35.7476806640625,
+			"width": 128.28594970703125,
+			"height": 56.65948486328125,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 0.5,
+			"strokeStyle": "solid",
+			"roughness": 0,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 1331034359,
+			"version": 137,
+			"versionNonce": 2121316023,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035105083,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					128.28594970703125,
+					-56.65948486328125
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		},
+		{
+			"id": "C0sy3Xpg-s5Gnc87lAnoq",
+			"type": "line",
+			"x": -212.6644287109375,
+			"y": 25.55078125,
+			"width": 45.83074951171875,
+			"height": 47.34820556640625,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 0.5,
+			"strokeStyle": "solid",
+			"roughness": 0,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"seed": 1246926521,
+			"version": 86,
+			"versionNonce": 2000085975,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1706035102645,
+			"link": null,
+			"locked": false,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					-45.83074951171875,
+					-47.34820556640625
+				]
+			],
+			"lastCommittedPoint": null,
+			"startBinding": null,
+			"endBinding": null,
+			"startArrowhead": null,
+			"endArrowhead": null
+		}
+	],
+	"appState": {
+		"theme": "dark",
+		"viewBackgroundColor": "#ffffff",
+		"currentItemStrokeColor": "#1e1e1e",
+		"currentItemBackgroundColor": "transparent",
+		"currentItemFillStyle": "solid",
+		"currentItemStrokeWidth": 0.5,
+		"currentItemStrokeStyle": "solid",
+		"currentItemRoughness": 0,
+		"currentItemOpacity": 100,
+		"currentItemFontFamily": 1,
+		"currentItemFontSize": 20,
+		"currentItemTextAlign": "left",
+		"currentItemStartArrowhead": null,
+		"currentItemEndArrowhead": "arrow",
+		"scrollX": 711,
+		"scrollY": 360.9765625,
+		"zoom": {
+			"value": 1
+		},
+		"currentItemRoundness": "round",
+		"gridSize": null,
+		"gridColor": {
+			"Bold": "#C9C9C9FF",
+			"Regular": "#EDEDEDFF"
+		},
+		"currentStrokeOptions": null,
+		"previousGridSize": null,
+		"frameRendering": {
+			"enabled": true,
+			"clip": true,
+			"name": true,
+			"outline": true
+		}
+	},
+	"files": {}
+}
+```
+%%
\ No newline at end of file
diff --git a/docs/excalidraw/Drawing 2024-01-22 11.18.38.excalidraw.md b/docs/excalidraw/Drawing 2024-01-22 11.18.38.excalidraw.md
new file mode 100644
index 0000000..2ce70ea
--- /dev/null
+++ b/docs/excalidraw/Drawing 2024-01-22 11.18.38.excalidraw.md	
@@ -0,0 +1,331 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠==
+
+
+# Text Elements
+l ^RCWgu8yn
+
+r ^ioDe3bEb
+
+value ^IRaq65B0
+
+hash(key)->index ^IbJj9bKd
+
+%%
+# Drawing
+```json
+{
+	"type": "excalidraw",
+	"version": 2,
+	"source": "https://github.com/zsviczian/obsidian-excalidraw-plugin/releases/tag/2.0.17",
+	"elements": [
+		{
+			"type": "rectangle",
+			"version": 38,
+			"versionNonce": 1400939928,
+			"isDeleted": false,
+			"id": "wgGv2I5Mv3b9REhmfRysL",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": -219.96893310546875,
+			"y": -160.53964233398438,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 434.0501708984375,
+			"height": 239.64236450195312,
+			"seed": 1376707736,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 3
+			},
+			"boundElements": [],
+			"updated": 1705915122923,
+			"link": null,
+			"locked": false
+		},
+		{
+			"type": "line",
+			"version": 88,
+			"versionNonce": 1926792424,
+			"isDeleted": false,
+			"id": "5-DTYhmTCi-lkEyyN5Ik8",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": -218.62945556640625,
+			"y": -41.552490234375,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 436.9171142578125,
+			"height": 1.129425048828125,
+			"seed": 1015558808,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"boundElements": [],
+			"updated": 1705915128660,
+			"link": null,
+			"locked": false,
+			"startBinding": null,
+			"endBinding": null,
+			"lastCommittedPoint": null,
+			"startArrowhead": null,
+			"endArrowhead": null,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					436.9171142578125,
+					-1.129425048828125
+				]
+			]
+		},
+		{
+			"type": "line",
+			"version": 80,
+			"versionNonce": 1979023512,
+			"isDeleted": false,
+			"id": "ywqog2oYcQPfVpRLKv6L-",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": 5.089599609375,
+			"y": -160.41998291015625,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 1.2037353515625,
+			"height": 120.68948364257812,
+			"seed": 1783080936,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": {
+				"type": 2
+			},
+			"boundElements": [],
+			"updated": 1705915134140,
+			"link": null,
+			"locked": false,
+			"startBinding": null,
+			"endBinding": null,
+			"lastCommittedPoint": null,
+			"startArrowhead": null,
+			"endArrowhead": null,
+			"points": [
+				[
+					0,
+					0
+				],
+				[
+					-1.2037353515625,
+					120.68948364257812
+				]
+			]
+		},
+		{
+			"type": "text",
+			"version": 2,
+			"versionNonce": 936925928,
+			"isDeleted": false,
+			"id": "RCWgu8yn",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": -101.7783203125,
+			"y": -92.49819946289062,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 5.2599945068359375,
+			"height": 25,
+			"seed": 296332696,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"boundElements": [],
+			"updated": 1705915135758,
+			"link": null,
+			"locked": false,
+			"fontSize": 20,
+			"fontFamily": 1,
+			"text": "l",
+			"rawText": "l",
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"containerId": null,
+			"originalText": "l",
+			"lineHeight": 1.25,
+			"baseline": 18
+		},
+		{
+			"type": "text",
+			"version": 2,
+			"versionNonce": 2096246424,
+			"isDeleted": false,
+			"id": "ioDe3bEb",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": 99.6004638671875,
+			"y": -102.03213500976562,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 8.599990844726562,
+			"height": 25,
+			"seed": 1767686376,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"boundElements": [],
+			"updated": 1705915137925,
+			"link": null,
+			"locked": false,
+			"fontSize": 20,
+			"fontFamily": 1,
+			"text": "r",
+			"rawText": "r",
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"containerId": null,
+			"originalText": "r",
+			"lineHeight": 1.25,
+			"baseline": 18
+		},
+		{
+			"type": "text",
+			"version": 30,
+			"versionNonce": 1557195240,
+			"isDeleted": false,
+			"id": "IRaq65B0",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"angle": 0,
+			"x": -30.814208984375,
+			"y": 9.98883056640625,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"width": 51.359954833984375,
+			"height": 25,
+			"seed": 377055384,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"boundElements": [],
+			"updated": 1705915143174,
+			"link": null,
+			"locked": false,
+			"fontSize": 20,
+			"fontFamily": 1,
+			"text": "value",
+			"rawText": "value",
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"containerId": null,
+			"originalText": "value",
+			"lineHeight": 1.25,
+			"baseline": 18
+		},
+		{
+			"id": "IbJj9bKd",
+			"type": "text",
+			"x": -144.08932359955736,
+			"y": -197.28850381874014,
+			"width": 154.27984619140625,
+			"height": 25,
+			"angle": 0,
+			"strokeColor": "#1e1e1e",
+			"backgroundColor": "transparent",
+			"fillStyle": "solid",
+			"strokeWidth": 2,
+			"strokeStyle": "solid",
+			"roughness": 1,
+			"opacity": 100,
+			"groupIds": [],
+			"frameId": null,
+			"roundness": null,
+			"seed": 2039595672,
+			"version": 17,
+			"versionNonce": 215778024,
+			"isDeleted": false,
+			"boundElements": null,
+			"updated": 1705915527505,
+			"link": null,
+			"locked": false,
+			"text": "hash(key)->index",
+			"rawText": "hash(key)->index",
+			"fontSize": 20,
+			"fontFamily": 1,
+			"textAlign": "left",
+			"verticalAlign": "top",
+			"baseline": 18,
+			"containerId": null,
+			"originalText": "hash(key)->index",
+			"lineHeight": 1.25
+		}
+	],
+	"appState": {
+		"theme": "light",
+		"viewBackgroundColor": "#ffffff",
+		"currentItemStrokeColor": "#1e1e1e",
+		"currentItemBackgroundColor": "transparent",
+		"currentItemFillStyle": "solid",
+		"currentItemStrokeWidth": 2,
+		"currentItemStrokeStyle": "solid",
+		"currentItemRoughness": 1,
+		"currentItemOpacity": 100,
+		"currentItemFontFamily": 1,
+		"currentItemFontSize": 20,
+		"currentItemTextAlign": "left",
+		"currentItemStartArrowhead": null,
+		"currentItemEndArrowhead": "arrow",
+		"scrollX": 348.5351854159636,
+		"scrollY": 246.27029245399405,
+		"zoom": {
+			"value": 2
+		},
+		"currentItemRoundness": "round",
+		"gridSize": null,
+		"gridColor": {
+			"Bold": "#C9C9C9FF",
+			"Regular": "#EDEDEDFF"
+		},
+		"currentStrokeOptions": null,
+		"previousGridSize": null,
+		"frameRendering": {
+			"enabled": true,
+			"clip": true,
+			"name": true,
+			"outline": true
+		}
+	},
+	"files": {}
+}
+```
+%%
\ No newline at end of file