Added logical proposition / algorithm theory notes

Terms added from ECS 122A, ECS 154A, MAT 108
jzkyu · Oct 4, 2023 · 1d87817 · 1d87817
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diff --git a/docs/Public/.DS_Store b/docs/Public/.DS_Store
diff --git a/docs/Public/Math/Axioms/cancellation.md b/docs/Public/Math/Axioms/cancellation.md
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+Let $m$, $n$, and $p$ be integers. If $m\cdot n=m\cdot p$ and $m\ne 0$, then:
+$$
+n=p
+$$
diff --git a/docs/Public/Math/Axioms/reflexivity.md b/docs/Public/Math/Axioms/reflexivity.md
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+If $m$ is an [[integer]], then:
+$$
+m=m
+$$
diff --git a/docs/Public/Math/Axioms/replacement.md b/docs/Public/Math/Axioms/replacement.md
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+If $m=n$, then $n$ can be substituted for $m$ in any statement without changing the meaning of that statement. 
diff --git a/docs/Public/Math/Axioms/symmetry.md b/docs/Public/Math/Axioms/symmetry.md
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+Let $m$ and $n$ be integers. If $m=n$, then:
+$$
+n=m
+$$
+
diff --git a/docs/Public/Math/Axioms/transivity.md b/docs/Public/Math/Axioms/transivity.md
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+Let $m$, $n$, and $p$ be integers. If $m=n$ and $n=p$, then:
+$$
+m=p
+$$
diff --git a/docs/Public/Math/Logic/and.md b/docs/Public/Math/Logic/and.md
diff --git a/docs/Public/Math/Logic/conjunction.md b/docs/Public/Math/Logic/conjunction.md
diff --git a/docs/Public/Math/Logic/definition.md b/docs/Public/Math/Logic/definition.md
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+A *definition* is a statement of the meaning of a term. *Definition* is represented as:
+$$
+:=
+$$
+It can be read as: "is defined as". 
diff --git a/docs/Public/Math/Logic/disjunction.md b/docs/Public/Math/Logic/disjunction.md
diff --git a/docs/Public/Math/Logic/existential quantification.md b/docs/Public/Math/Logic/existential quantification.md
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+*Existential quantification* states that there at is at least one value in the [[domain]] of $x$ that will make the statement true. *Existential quantification* is represented as:
+$$
+\exists
+$$
+It can be read as: "there exists". 
diff --git a/docs/Public/Math/Logic/implication.md b/docs/Public/Math/Logic/implication.md
diff --git a/docs/Public/Math/Logic/negation.md b/docs/Public/Math/Logic/negation.md
diff --git a/docs/Public/Math/Logic/proposition.md b/docs/Public/Math/Logic/proposition.md
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+A *proposition* is a statement or assertion, that is either true or false. [^1]
+
+[^1]: https://brilliant.org/wiki/propositional-logic/
diff --git a/docs/Public/Math/Logic/propositional logic.md b/docs/Public/Math/Logic/propositional logic.md
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+*Propositional logic* is a branch of mathematics that studies the logical relationships of [[proposition]]. [^1]
+
+[^1]: https://brilliant.org/wiki/propositional-logic/
diff --git a/docs/Public/Math/Logic/universal quantification.md b/docs/Public/Math/Logic/universal quantification.md
diff --git a/docs/Public/Math/Proofs/proof by induction.md b/docs/Public/Math/Proofs/proof by induction.md
diff --git a/docs/Public/Math/Theory/limit lemma.md b/docs/Public/Math/Theory/limit lemma.md
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+*See:* [[theta notation]], [[Big-O notation]], [[omega notation]]
+
+Given $f(n)$ and $g(n)$, non-decreasing positive functions, then for $$\lim_{x \to \infty} \frac{f(x)}{g(x)}=L$$
+1. If $L=0$, then $f(n)=O(g(n))$
+2. If $L=c$, then $f(n)= \theta(g(n))$ for some constant $c$
+3. If $L= \infty$, then $f(n)= \Omega(g(n))$
+
diff --git a/docs/Public/Math/base.md b/docs/Public/Math/base.md
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+The number of unique digits, including zero, used to represent numbers. [^1]
+
+[^1]: https://en.wikipedia.org/wiki/Radix
diff --git a/docs/Public/Math/binary.md b/docs/Public/Math/binary.md
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+*Binary* is a [[numeral system]] which only uses the digits $0$ and $1$. It is [[base]] 2.
+
+Circuits and logic in computing can be purely represented in *binary*. in general, it can be used to represent two things or states. 
+
+
+
diff --git a/docs/Public/Math/boolean.md b/docs/Public/Math/boolean.md
@@ -0,0 +1,3 @@
+*See:* [[boolean algebra]]
+
+*Boolean* refers to variables which can only be one of two values: true or false. *Boolean* variables are usually denoted as a $1$ or $0$ as [[binary]].
diff --git a/docs/Public/Math/calculus.md b/docs/Public/Math/calculus.md
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+*Calculus* is a branch of mathematics that studies change. 
+
+It deals with [[continuous]] functions and using the properties of the [[derivative]] and [[integral]]. 
+
+*Calculus* is a powerful tool that can be used to model real world phenomena. It can reveal new characteristics in systems and models with respect to time. 
diff --git a/docs/Public/Math/continuous.md b/docs/Public/Math/continuous.md
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+In simple terms, a function or variable is continuous if its graph can be drawn in one stroke with a pencil. [^1]
+
+A function is discontinuous if there are breaks in the graph. If you have to lift your pencil, it means you made a very small movement but the function changed significantly. [^2]
+
+*Continuous* deals with [[real numbers]] and [[complex numbers]]. It is associated with measuring physical phenomena which *continuously* change, such as mechanical, electrical, and hydraulic. [^3]
+
+[^1]: https://www.zhenkaiweng.com/continuity/
+[^2]: https://simple.wikipedia.org/wiki/Continuous_function#:~:text=A%20mathematical%20function%20is%20called,uninterrupted%20line%20(or%20curve).
+[^3]: https://www.toppr.com/guides/physics/electronics/analog-computer-definition-and-types-of-analog-computers/#:~:text=An%20Analog%20computer%20is%20a,work%20on%20an%20analog%20signal.
diff --git a/docs/Public/Math/derivative.md b/docs/Public/Math/derivative.md
diff --git a/docs/Public/Math/discrete.md b/docs/Public/Math/discrete.md
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+*Discrete* refers to mathematical structures that are countable or otherwise distinct and separate. [^1] 
+
+In other words, *discrete* structures can only take certain values. For example, the number of students in class; you can't have half a student. [^2]
+
+Examples include integers, graphs, and statements in logic. [^3]
+
+[^1]: https://brilliant.org/wiki/discrete-mathematics/#:~:text=Discrete%20mathematics%20is%20the%20study,can%20be%20finite%20or%20infinite.
+[^2]: https://www.mathsisfun.com/definitions/discrete-data.html
+[^3]: https://en.wikipedia.org/wiki/Discrete_mathematics
diff --git a/docs/Public/Math/equal.md b/docs/Public/Math/equal.md
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+The symbol $=$ means *equal*. To say $m=n$ means that $m$ and $n$ are the same number. [^1]
+
+[^1]: https://matthbeck.github.io/papers/aop.noprint.pdf
diff --git a/docs/Public/Math/hexadecimal.md b/docs/Public/Math/hexadecimal.md
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+*Hexadecimal* is a [[numeral system]] that is [[base]] 16; it uses the following set of symbols for 
diff --git a/docs/Public/Math/integer.md b/docs/Public/Math/integer.md
@@ -1 +1 @@
-An *integer* is a whole number, including 0 and negative whole numbers. 
+An *integer* is a whole number, including 0 and negative whole numbers. The [[set]] of *integers* is denoted by $\mathbb{Z}$.
diff --git a/docs/Public/Math/integral.md b/docs/Public/Math/integral.md
diff --git a/docs/Public/Math/natural.md b/docs/Public/Math/natural.md
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+The *natural* numbers refers to the positive countable ([[discrete]]) numbers, sometimes including $0$. The [[set]] of *natural* numbers is denoted by $\mathbb{N}$.
diff --git a/docs/Public/Math/numeral system.md b/docs/Public/Math/numeral system.md
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+A *numeral system* is a writing system for expressing numbers. Given a [[set]], a *numerical system* sets the rules for how it will be represented as numbers. [^1]
+
+Common *numeral systems* are:
+- [[binary]] 
+	- The basis for all computing systems and operations; turning on and off circuits. 
+- [[octal]]
+- [[decimal]]
+	- 
+- [[hexadecimal]]
+	- Used in assembly / machine code, often to represent memory addresses
+
+[^1]: https://byjus.com/maths/number-system/#:~:text=What%20is%20Number%20System%20and,system%2C%20and%20hexadecimal%20number%20system.
diff --git a/docs/Public/Math/operand.md b/docs/Public/Math/operand.md
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+An *operand* is an element that can be manipulated in an [[operation]]. 
diff --git a/docs/Public/Math/operation.md b/docs/Public/Math/operation.md
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+An *operation* refers to a process, usually in math. An *operation* calculates a value using an [[operand]] and [[operator]]. [^1]
+
+[^1]: https://www.splashlearn.com/math-vocabulary/addition/operation
diff --git a/docs/Public/Math/operator.md b/docs/Public/Math/operator.md
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+An *operator* is the symbol used to carry out an [[operation]] between [[operand]]. [^1]
+
+[^1]: https://math.stackexchange.com/questions/920699/operator-vs-operation-vs-function-vs-procedure-vs-algorithm#:~:text=One%20can%20even%20see%20books,S%2CS)%2D%3ES.
diff --git a/docs/Public/Math/series.md b/docs/Public/Math/series.md
diff --git a/docs/Public/Math/unary.md b/docs/Public/Math/unary.md
diff --git a/docs/Public/Media/big_o.png b/docs/Public/Media/big_o.png
diff --git a/docs/Public/Media/omega.png b/docs/Public/Media/omega.png
diff --git a/docs/Public/Media/theta.png b/docs/Public/Media/theta.png
diff --git a/docs/Public/Media/transistor.png b/docs/Public/Media/transistor.png
diff --git a/docs/Public/Media/transistor_connection.png b/docs/Public/Media/transistor_connection.png
diff --git a/docs/Public/Media/tree.png b/docs/Public/Media/tree.png
diff --git a/docs/Public/Software/Computer Architecture/analog.md b/docs/Public/Software/Computer Architecture/analog.md
diff --git a/docs/Public/Software/Computer Architecture/digital.md b/docs/Public/Software/Computer Architecture/digital.md
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+*Digital* refers to electronic technology that use [[discrete]] values, generally zero and one, to generate, store, and process [[data]]. [^1]
+
+[^1]: https://www.techopedia.com/definition/604/digital-definition
diff --git a/docs/Public/Software/Computer Architecture/semiconductor.md b/docs/Public/Software/Computer Architecture/semiconductor.md
diff --git a/docs/Public/Software/Computer Architecture/short circuit.md b/docs/Public/Software/Computer Architecture/short circuit.md
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+A *short circuit* happens when both power and ground are connected. 
diff --git a/docs/Public/Software/Computer Architecture/transistor.md b/docs/Public/Software/Computer Architecture/transistor.md
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+A *transistor* is an electrical switch in circuits. There are no moving parts. 
+
+MOS = Metal Oxide [[semiconductor]]
+There are two types of transistors: *nMOS* and *pMOS*
+
+![[transistor.png]]
+
+How they work: 
+Current flows when there is an electrical difference i.e. positive to negative. When a positive voltage is applied to the gate, it attracts electrons from the substrate, making the top part of the substrate behave like it is negatively charged. Now that there is an electrical difference, the source and drain are connected, and current flows. 
+
+![[transistor_connection.png]]
diff --git a/docs/Public/Software/Theory/Big-O notation.md b/docs/Public/Software/Theory/Big-O notation.md
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+*Big-O notation* represents the upper bound of the running time of an [[algorithm]]. It gives the worst case complexity of an algorithm. [^1]
+
+```
+O(g(n)) = { f(n): there exist positive constants c and n0
+            such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 }
+```
+![[big_o.png]]
+
+[^1]: https://www.programiz.com/dsa/asymptotic-notations
diff --git a/docs/Public/Software/Theory/abstraction.md b/docs/Public/Software/Theory/abstraction.md
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+*Abstraction* is the principle of simplifying a complex system into a basic model. By *abstracting* the details, the new model makes it easier to work with the complex system. [^1]
+
+A car has many systems: Brakes, transmission, suspension system, battery, etc. But we don't need to understand how each of those systems work to be able to drive a car; we just need to use the interface, i.e. the steering wheel, accelerator, and brake pedal, to use the car. We are not concerned with the inner details of the car when driving. [^1]
+
+[^1]: https://www.freecodecamp.org/news/what-is-abstraction-in-programming/
diff --git a/docs/Public/Software/Theory/algorithm analysis.md b/docs/Public/Software/Theory/algorithm analysis.md
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+*See:* [[theta notation]], [[Big-O notation]], [[omega notation]]
+
+*Algorithm analysis* generally refers to the "complexity analysis" of an algorithm: how the characteristics of the [[algorithm]] changes with respect to changes in parameters.
+
+In other words, how does the input size of an algorithm affect: [^1]
+- [[runtime]] (instructional steps)
+- space (in memory)
+- N / W (data transferred / network consumption)
+- power consumption (for battery life)
+- CPU registers (physical space used on the processor)
+
+*Algorithm analysis* is useful to measure the efficiency of an algorithm, since it is machine-independent. 
+
+[^1]: https://www.youtube.com/watch?v=xGYsEqe9Vl0
diff --git a/docs/Public/Software/Theory/algorithm.md b/docs/Public/Software/Theory/algorithm.md
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+*See:* [[algorithm analysis]]
+
+An *algorithm* is a procedure that takes an input and produces an output. [^1]
+
+[^1]: https://dahlan.unimal.ac.id/files/ebooks/2009%20Introduction%20to%20Algorithms%20Third%20Ed.pdf
diff --git a/docs/Public/Software/Theory/data structure.md b/docs/Public/Software/Theory/data structure.md
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+A *data structure* is a way to store and organize [[data]]. You can access and modify the data in a *data structure*. [^1]
+
+[^1]: https://dahlan.unimal.ac.id/files/ebooks/2009%20Introduction%20to%20Algorithms%20Third%20Ed.pdf
diff --git a/docs/Public/Software/Theory/data.md b/docs/Public/Software/Theory/data.md
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+*Data* is distinct pieces of information, quantified or qualified by some identifier which gives it meaning. [^1] [^2]
+
+Raw *data* is information in a pure form like characters and numbers. We process raw *data* through many methods to then produce useful *data* that gives us key insights. 
+
+[^1]: https://en.wikipedia.org/wiki/Data_(computer_science)
+[^2]: https://www.geeksforgeeks.org/what-is-data/
diff --git a/docs/Public/Software/Theory/graph.md b/docs/Public/Software/Theory/graph.md
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+A *graph* is a data structure consisting of vertices and edges. [^1]
+
+A vertex is often referred to as a [[node]].[^1]
+
+More formally, a *graph* is composed of a [[set]] of $vertices(V)$ and a set of $edges(E)$. The graph is denoted by $G(E, V)$.[^1]
+
+[^1]: https://www.geeksforgeeks.org/graph-data-structure-and-algorithms/
diff --git a/docs/Public/Software/Theory/heap.md b/docs/Public/Software/Theory/heap.md
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+A *heap* is a 
diff --git a/docs/Public/Software/Theory/node.md b/docs/Public/Software/Theory/node.md
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+A *node* is a basic unit of a [[data structure]]. A *Node* contains data and also may link to other *nodes*.[^1]
+
+[^1]: https://www.wikiwand.com/en/Node_(computer_science)
diff --git a/docs/Public/Software/Theory/omega notation.md b/docs/Public/Software/Theory/omega notation.md
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+*Omega notation* represents the lower bound of the running time of an [[algorithm]]. It is the base case complexity of an algorithm. [^1]
+
+```
+Ω(g(n)) = { f(n): there exist positive constants c and n0 
+            such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0 }
+```
+
+![[omega.png]]
+
+[^1]: https://www.programiz.com/dsa/asymptotic-notations
diff --git a/docs/Public/Software/Theory/root.md b/docs/Public/Software/Theory/root.md
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+A *root* is the first node in a [[tree]] [[data structure]]. 
diff --git a/docs/Public/Software/Theory/runtime.md b/docs/Public/Software/Theory/runtime.md
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+*Runtime* is the period of time while a program is running on a computer. 
+
+Common algorithmic runtimes from fastest to slowest are: [^1]
+- Constant: $O(1)$
+- Logarithmic: $O(log n)$
+- Linear: $O(n)$
+- Polynomial: $O(n^2)$
+- Exponential: $O(2^n)$
+- Factorial: $O(n!)$
+
+[^1]: https://www.codecademy.com/learn/cspath-asymptotic-notation/modules/cspath-asymptotic-notation/cheatsheet
diff --git a/docs/Public/Software/Theory/theta notation.md b/docs/Public/Software/Theory/theta notation.md
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+*Theta notation* encloses the function from above and below. Since it represents the upper and the lower bound of the running time of an [[algorithm]], it is used for analyzing the average case complexity of an algorithm. [^1]
+
+```
+Θ(g(n)) = { f(n): there exist positive constants c1, c2 and n0
+            such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0 }
+```
+
+![[theta.png]]
+
+[^1]: https://www.programiz.com/dsa/asymptotic-notations
diff --git a/docs/Public/Software/Theory/tree.md b/docs/Public/Software/Theory/tree.md
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+A *tree* is a type of [[graph]] [[data structure]] that has a hierarchical "tree" structure. A [[root]] [[node]] "branches" off to its connected children nodes, and those nodes act as a parent node to other children nodes. You can traverse through the *tree* from the root to any node in a unique path in one direction. 
+
+*Trees* can be used to store data that has an inherent hierarchical structure i.e. directories, files, and folders in file management systems. [^1]
+
+*Trees* are easy to sort and search through using algorithms. 
+
+![[tree.png]]
+
+[^1]: https://computersciencewiki.org/index.php/Tree
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		If $m=n$, then $n$ can be substituted for $m$ in any statement without changing the meaning of that statement.
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		A proposition is a statement or assertion, that is either true or false. [^1]

		[^1]: https://brilliant.org/wiki/propositional-logic/
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		Propositional logic is a branch of mathematics that studies the logical relationships of [[proposition]]. [^1]

		[^1]: https://brilliant.org/wiki/propositional-logic/
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		The number of unique digits, including zero, used to represent numbers. [^1]

		[^1]: https://en.wikipedia.org/wiki/Radix
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		Binary is a [[numeral system]] which only uses the digits $0$ and $1$. It is [[base]] 2.

		Circuits and logic in computing can be purely represented in binary. in general, it can be used to represent two things or states.
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		See: [[boolean algebra]]

		Boolean refers to variables which can only be one of two values: true or false. Boolean variables are usually denoted as a $1$ or $0$ as [[binary]].
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		The symbol $=$ means equal. To say $m=n$ means that $m$ and $n$ are the same number. [^1]

		[^1]: https://matthbeck.github.io/papers/aop.noprint.pdf
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		Hexadecimal is a [[numeral system]] that is [[base]] 16; it uses the following set of symbols for
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		An integer is a whole number, including 0 and negative whole numbers.
		An integer is a whole number, including 0 and negative whole numbers. The [[set]] of integers is denoted by $\mathbb{Z}$.
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		The natural numbers refers to the positive countable ([[discrete]]) numbers, sometimes including $0$. The [[set]] of natural numbers is denoted by $\mathbb{N}$.