Tag: Math

What is the Gist of “Directed Graphs,” “Undirected Graphs,” “Oriented Graphs,” and “Networks” (Graph Theory)?

GraphsDirected

“The edges* of graphs* may… be imbued with directedness. A normal graph in which edges are undirected is said to be undirected. Otherwise, if arrows may be placed on one or both endpoints of the edges of a graph to indicate directedness, the graph is said to be directed. A directed graph in which each edge is given a unique direction (i.e., edges may not be bidirected and point in both directions at once) is called an oriented graph. A graph or directed graph together with a function which assigns a positive real number to each edge (i.e., an oriented edge-labeled graph) is known as a network.” (“Graph,” WolframMathWorld, retrieved 6/25/2020, including the picture).

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*What is the Gist of a “Graph” (Graph Theory)?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Simple Graphs,” “Multigraphs,” and Pseudographs” (Graph Theory)?

GraphsSimple

“Graphs* come in a wide variety of different sorts**. The most common type is graphs in which at most one edge* (i.e., either one edge or no edges) may connect any two vertices*. Such graphs are called simple graphs. If multiple edges are allowed between vertices, the graph is known as a multigraph***. Vertices are usually not allowed to be self-connected, but this restriction is sometimes relaxed to allow such “graph loops.” A graph that may contain multiple edges and graph loops is called a pseudograph” (“Graph,” WolframMathWorld, retrieved 6/25/2020, including the picture).

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*What is the Gist of a “Graph” (Graph Theory)?

**The Wikipedia page called “Graphs” contains a section describing many other types beyond those shown here (retrieved 6/25/2020).

***This is the general rule. However, according to “Multigraph,” from WolframMathWorld, there can be a lot of ambiguity in what is and isn’t required for a graph to be a multigraph.

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Graph” (Graph Theory)?

“The word “graph” has (at least) two meanings in mathematics.

“In elementary mathematics, “graph” refers to a function graph or “graph of a function,” i.e., a plot.

“In a mathematician’s terminology, [especially in graph theory], a graph is a collection of points and lines connecting some (possibly empty) subset* of them. The points of a graph are most commonly known as graph vertices, but may also be called “nodes” or simply “points.” Similarly, the lines connecting the vertices of a graph are most commonly known as graph edges, but may also be called “arcs” or “lines.”” (“Graph,” WolframMathWorld, retrieved 6/25/2020)

Graphs seem to be used to show how different parts of a thing or things are connected with each other.

If interested in getting a better idea of what graphs are, a good starting point would be to read “Graph,” from WolframMathWorld, (retrieved 6/25/2020) and, after the other link, “Graph (discrete mathematics),” from Wikipedia (retrieved 6/25/2020), especially the section “Properties of graphs.”

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*What is the Gist of a “Subset”?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Cartesian Product”?

“The Cartesian product of two sets A and B… is defined to be the set of all points (a, b) where a [is in set] A and b [is in set] B.” (“Cartesian Product,” WolframMathWorld). For example, if A = {1,2,3} and B = {4,5}, A x B = {(1,4), (1,5), (2,4), (2,5), (3,4), (3,5)}

Note, the Cartesian product is sometimes called “the product set, set direct product, or cross product”(ibid). That said, the cross product usually seems to apply to vectors. How exactly the two are related is unclear to me at the moment.

Cartesian coordinates seem to be related to the Cartesian product because Euclidean Spaces,* in which you can use Cartesian coordinates, are created when having the set of real numbers do the Cartesian product with itself (ibid).

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*What is the Gist of “Euclidean Space”?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of the “Complement of a Set”?

“In set theory, the complement of a set A refers to elements not in A.”

If A is a subset* of a given set U, then the absolute complement of A is the set of elements that are in U but not in A. Note that U may be defined only implicitly.

If A and B are both subsets* of a given set U, then “the relative complement of A with respect to a set B, also [called] the set difference of B and A, written B \ A, is the set of elements [that are] in B but not in A.“

Quotes and information taken from “Complement (set theory),” Wikipedia, retrieved 6/17/2020, emphasis added. Note that, as of retrieving this information, there are some pictures there that could be useful in understanding this.

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*What is the Gist of a “Subset”?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Family of Sets”?

“A collection F of subsets* of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

“The term “collection” is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member” (“Family of sets,” Wikipedia, retrieved 7/17/2020).

Note that the set S is a subset* of itself, so can be in F.

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*What is the Gist of a “Subset”?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Subset”?

“A set A is a subset of a set B… [if] all elements of A are also elements of B.” (“Subset,” Wikipedia, retrieved 6/17/2020, emphasis added)

For some more terminology, here’s the complete quote if you are interested:

“In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B. That is, all elements of A are also elements of B. A and B may be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion or sometimes containment.” (Ibid.)

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Mapping” (Mathematics)?

“Mapping… often refers to the action of applying a function to the elements of its domain. This terminology is not completely fixed, as these terms are generally not formally defined.” (“Map (mathematics),” Wikipedia, retrieved 6/15/2020)

(Using “Ctrl-f” to search for “mapping” in the article “Function (mathematics),” Wikipedia, searched 6/15/2020, was also helpful)

As a noun (a “map” or a “mapping”), the definition is more confusing. The opening paragraphs of the “Map (mathematics)” article mentioned earlier seem to have a good, if difficult to read, summary. The gist is that saying “a map” or a “mapping” can often, but not always, be considered synonymous with saying “a function.”

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of a “Manifold”?

“In mathematics, a manifold is a topological space that locally resembles Euclidean space* near each point” (“Manifold,” Wikipedia, retrieved 6/10/2020). In other words, if you zoom in close enough to any point on a manifold, you won’t be able to tell the difference between what you see and a Euclidean space*. For example, if you zoom in close enough to a circle, it will look almost exactly like a straight line (ibid). This also works for most of a figure 8, but if you focus on the middle of the figure 8, no amount of zooming in will ever make it look like a single straight line (ibid).

There exists something called “Manifolds with a boundary.” “A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an [n-dimensional ‘manifold with a boundary’] is an (n−1)-[dimensional] manifold… A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold (ibid, italics added).”

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*What is the Gist of “Euclidean Space”?

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. By definition, none of these posts address every aspect of a topic. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).

What is the Gist of “Euclidean Space”?

A 3-dimensional Euclidean space (which is normally what is being referred to if talking about a Euclidean space) is basically the space in which we live (excluding relativistic effects), where moving and rotating things don’t change them and you can use Cartesian coordinates to say where things are. The 2-dimensional Euclidean space is basically a plane, and a 1-dimensional Euclidean space is basically a straight line. Other dimensions of Euclidean space are allowed.

Source: “Euclidean space,” Wikipedia, retrieved 6/10/2020

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Disclaimer:

I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).