Sets, FSM

- Sets, set relation terms
- State machines, computer architecture terms

docs/Public/Math/Sets/cartesian product.md

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+Let $A$ and $B$ be some [[set]]. The set of all ordered pairs $(a, b)$ such that $a\in A$ and $b \in B$ is called the *cartesian product*. 
+$$
+A\times B = \{(a, b)| a\in A, b\in B\}
+$$
+Example:
+Let $A = \{m, p\}$, $B = \{5, 7, 9\}$. Then, 
+$$
+A\times B = \{(m, 5), (m, 7), (m, 9), (p, 5), (p, 7), (p, 9)\}
+$$

docs/Public/Math/Sets/empty set.md

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+The *empty set* is the [[set]] with no elements. There is no such $x\in \emptyset$.

docs/Public/Math/Sets/intersection.md

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+The *intersection* of two sets contains only the elements contained in both [[set]].
+
+The *intersection* of $A$ and $B$ is:
+$$
+A \cup B = \{ x | x\in A \land x\in B\}
+$$

docs/Public/Math/Sets/power set.md

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+If $A$ is a set, then the *power set* of $A$ is the set whose elements are the [[subset]] of A, and it is denoted by $P(A)$. 
+$$
+P(A) = \{ B | B \in A \}
+$$
+
+

docs/Public/Math/Sets/rational.md

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+A *rational* is a number that can be expressed as the fraction of two [[integer]], as in
+$$
+\frac{p}{q}
+$$

docs/Public/Math/Sets/relation.md

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+A *relation* $\mathbb{R}$ from $A$ to $B$ is a [[subset]] of $A\times B$. 
+
+A *relation* $\mathbb{R}$ from $A$ to $A$ is also called a *relation* on $A$. 

docs/Public/Math/Sets/set difference.md

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+The *set difference* of $A$ and $B$ is:
+$$
+A - B = \{ x | x\in A \land x\not \in B \}
+$$

docs/Public/Math/Sets/set equality.md

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+$A$ and $B$ have *set equality*, $A=B$ if:
+$$
+A\subseteq B \text{ and } B\subseteq A
+$$

docs/Public/Math/Sets/subset.md

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+A *subset* is a [[set]] which all the elements are contained in another set. 
+
+$A$ is a *subset* of $B$ means:
+$$
+\text{If }x\in A, \text{ then } x\in B
+$$
+We write:
+$$
+a\subseteq B
+$$

docs/Public/Math/Sets/tuple.md

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+A *tuple* is an ordered [[set]] with $n$ elements. 

docs/Public/Math/Sets/union.md

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+The *union* of two sets contains all the elements contained in either [[set]].
+
+The *union* of $A$ and $B$ is:
+$$
+A \cup B = \{ x | x\in A \lor x\in B\}
+$$

docs/Public/Software/Computer Architecture/D latch.md

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+A *D latch* is type of [[latch]] that has one input: Data ($D$). The output of the *D latch* is the same as the $D$ input as long as the [[clock]] signal is high. [^1]
+
+[^1]: https://www.geeksforgeeks.org/latches-in-digital-logic/#

docs/Public/Software/Computer Architecture/Mealy state machine.md

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+A *Mealy state machine* is a [[finite state machine]] whose current outputs are dependent on the current and previous state. [^1]
+
+In other words, the output is built on the edges. 
+[^1]: https://www.geeksforgeeks.org/mealy-and-moore-machines-in-toc/#

docs/Public/Software/Computer Architecture/Moore state machine.md

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+A *Moore state machine* is a [[finite state machine]] whose current outputs are dependent only on the current state. [^1]
+
+In other words, the output will be 'inside' the state. 
+
+[^1]: https://www.geeksforgeeks.org/mealy-and-moore-machines-in-toc/#

docs/Public/Software/Computer Architecture/SR latch.md

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+A *SR latch* is a type of [[latch]] which has two inputs: set ($S$) and reset ($R$). The $S$ input sets the output to $1$, while the $R$ input sets the output to $0$. [^1]
+
+[^1]: https://www.geeksforgeeks.org/latches-in-digital-logic/#

docs/Public/Software/Computer Architecture/cache.md

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+A *cache* is a temporary [[data]] store that holds data so future requests for that data can be serve faster. 

docs/Public/Software/Computer Architecture/clock.md

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+The system *clock* is used as an input to the [[sequential circuit]], alternating between $1$ and $0$ at regular intervals (in a *clock* cycle), to control when the state of a circuit changes. [^1]
+
+[^1]: https://courses.cs.vt.edu/~cs1104/SystemsOrg/COrgn_9.htm#:~:text=All%20the%20circuits%20in%20the,state%20of%20a%20circuit%20changes.

docs/Public/Software/Computer Architecture/comparator.md

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+A *comparator* takes in two binary inputs, and checks to see if the bits of each input are the same value at each position. 

docs/Public/Software/Computer Architecture/finite state machine.md

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+A *finite state machine* is a model that describes systems with a finite number of states and transitions to represent their behavior and logic. [^1]
+
+There are different kinds of *finite state machines*:
+- [[Moore state machine]]
+- [[Mealy state machine]]
+
+[^1]: https://www.spiceworks.com/tech/tech-general/articles/what-is-fsm/

docs/Public/Software/Computer Architecture/flip flop.md

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+A *flip flop* is a [[latch]] that is controlled by a [[clock]]. 

docs/Public/Software/Computer Architecture/latch.md

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+A *latch* is a digital circuit that stores a single bit of information and holds its value until it is update by new input signals. [^1]
+
+There are several types of latches, including:
+- [[SR latch]]
+- [[D latch]]
+- 
+
+[^1]: https://www.geeksforgeeks.org/latches-in-digital-logic/#

docs/Public/Software/Theory/dynamic programming.md

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+*Dynamic programming* is optimization over [[recursion]]. Whenever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using *dynamic programming*. [^1]
+
+The idea is to store the results of the sub-problems, so we don't have to recompute them. 
+
+[^1]: https://www.geeksforgeeks.org/dynamic-programming/#

docs/Public/Software/Theory/memoization.md

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+*Memoization* is an optimization technique that stores computation results in [[cache]], and retrieving that information from the cache the next time it's needed instead of computing it again. [^1]
+
+[^1]: https://www.freecodecamp.org/news/memoization-in-javascript-and-react/