Math – Ruficalix

What is the Gist of “Rings” (Abstract Algebra)?

In general, a ring can be thought of as a set^[1] that has been combined with two binary operations^[2] and follows certain rules regarding what happens when various elements of the set are combined with each other using the two binary operations.

More specifically, to get a ring, start with an Abelian group^[3] whose operation is called addition. Now, pair it with a second binary operation called multiplication. This second binary operation, multiplication, must be associative^[4] and must be distributive^[5] over the addition operation. The Wikipedia article “Ring (mathematics)” says that multiplication must also have an identity^[6], but acknowledges that some authors do not require rings to have a multiplicative identity. For example, Abstract Algebra: Theory and Applications, by Thomas W. Judson, specifically calls a ring with a multiplicative identity a “ring with unity” or a “ring… with identity.”

The addition operation is often denoted with a plus sign (“+”) while the multiplication operation is often denoted with a multiplication sign (“*”) or no sign.

Sources: “Ring (mathematics),” Wikipedia, retrieved 12/15/2020, and Abstract Algebra: Theory and Applications, by Thomas W. Judson, August 9, 2016 edition.

[1] See “What is the Gist of a “Set” (Mathematics)?”
[2] See “What is the Gist of a “Binary Operation”?”
[3] See “What is the Gist of an “Abelian Group”?”
[4] See “What is the Gist of “Associativity” (Mathematics)?”
[5] See “What is the Gist of the “Distributive Property” (Mathematics)?”
[6] See “What is the Gist of “Identity” (Mathematics)?”

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 :).

What is the Gist of the “Distributive Property” (Mathematics)?

“Given a set^[1] S and two binary operators^[2] ∗ and + on S… :

“[the operation ∗] is left-distributive over + if, given any elements x, y and z of S, [we find that] $x*(y+z)=(x*y)+(x*z),$
“[the operation ∗] is right-distributive over + if, given any elements x, y, and z of S, [we find that] $(y+z)*x=(y*x)+(z*x),$ and
“[the operation ∗] is distributive over + if it is left- and right-distributive.” (“Distributive Property,” Wikipedia, retrieved 12/15/2020)

[1] See “What is the Gist of a “Set” (Mathematics)?”
[2] See “What is the Gist of a “Binary Operation”?”

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What is the Gist of an “Abelian Group”?

“In mathematics, an abelian group, also called a commutative group, is a group^[1] in which the result of applying the group operation to [any] two group elements does not depend on the order in which they are written. That is, the group operation is commutative^[2]” (“Abelian Group,” Wikipedia, retrieved 12/15/2020, bold added for emphasis). One of the best known Abelian groups is the set of real numbers paired with the binary operation^[3] of addition (ibid).

“A group in which the group operation is not commutative is called a ‘non-abelian group’ or ‘non-commutative group'” (ibid).

The group operation of an Abelian group is often denoted with a plus sign (“+“), as opposed to a multiplication sign (“*”) or no sign (ibid). Abelian groups are also sometimes referred to as abelian groups (note the lowercase “a”) (ibid).

[1] See “What is the Gist of a “Group” (Group Theory)?”
[2] See “What is the Gist of “Commutativity” (Mathematics)?”
[3] See “What is the Gist of a “Binary Operation”?”

Disclaimer:

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

“In mathematics, a binary operation^[1] is commutative if changing the order of the operands^[1] does not change the result… [Commutativity is often known] as the name of the property that says “3 + 4 = 4 + 3” or “2 × 5 = 5 × 2,” [although it] can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, “3 − 5 ≠ 5 − 3”); such operations are not commutative, and so [can be] referred to as noncommutative operations.” (“Commutative Property,” Wikipedia, retrieved 12/15/2020, bold added for emphasis)

“If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to commute under that operation.” (ibid)

The Wikipedia article also has some delightful example of commutativity and non-commutativity in every day life that could be useful in getting the gist of the idea: “Commutative operations in everyday life” and “Noncommutative operations in daily life” (retrieved 12/15/2020).

Note: commutativity means something different in propositional logic, and commutativity is not the same thing as associativity^[2] (ibid)

[1] See “What is the Gist of a “Binary Operation”?”
[2] See “What is the Gist of “Associativity” (Mathematics)?”

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What is the Gist of the Different Symbols Used to Denote a Group in Mathematics?

(Note: this post is not going to be as simple as most other posts due to time constraints. Instead, the intention of this post is to provide a starting place to understand the different names, like D4 and C5, for groups)

In the attached doc, I briefly cover the following:

Z (The Integers)
Z/nZ (The Integers Modulo n)
Z_n (The Integers Modulo n (in Group Theory))
C (Cyclic group)
C_n (Cyclic group of degree/order n)
S (Symmetric Group)
S_n (Finite Symmetric Group)
A_n (Alternating Groups)
D_n (Dihedral Group).

Names-of-Groups-Doc Download

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What is the Gist of an “Isomorphism”?

An isomorphism maps a mathematical structure to a mathematical structure in a way that preserves some part of the structure. It must also be possible to reverse an isomorphism with an inverse mapping.^[1]

“Two mathematical structures are isomorphic if an isomorphism exists between them…

“An automorphism is an isomorphism from a structure to itself”

“In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:

“An isometry^[2] is an isomorphism of metric spaces.
“A homeomorphism is an isomorphism of topological spaces.
“A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
“A permutation is an automorphism of a set.
“In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.

“In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other. Equality is when two objects are exactly the same, and everything that’s true about one object is true about the other, while an isomorphism implies everything that’s true about a designated part of one object’s structure is true about the other’s.” “In mathematical jargon, one says that two objects are the same up to an isomorphism.”

Information from “Isomorphism,” Wikipedia, retrieved 11/7/2020

[1] What is the Gist of an “Inverse Function”?

[2] What is the Gist of “Isometry” (Mathematics)?

Disclaimer:

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

“Given a metric space^[1] (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the… elements in the new metric space is equal to the distance between the elements in the original metric space.”

For example, “any reflection, translation and rotation is… [an] isometry on Euclidean spaces.^[2]“

Notes:

Isometries are also called a congruence, or a congruent transformation
“An isometry is automatically injective^[3]“
Isometries are “usually assumed to be bijective.^[3]”
If an isometry is bijective, it can also be called a “global isometry, isometric isomorphism or congruence mapping.” Reflection, translations, and rotations are global isometries on Euclidean spaces.
“Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y.”
“An isometry is an isomorphism^[4] of metric spaces.”*

Source of quotes/information: “Isometry,” Wikipedia, retrieved 11/7/2020, emphasis added to first paragraph

* The source of this line is different from the rest: “Isomorphism,” Wikipedia, retrieved 11/7/2020.

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What is the Gist of an “Inverse Function”?

“In mathematics, an inverse function (or anti-function) is a function that “reverses” another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. The inverse function of f is also denoted as $f^{-1}$ …

“Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y, there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.” In other words, f must be a bijection^[1] to be invertible.

Notes:

The inverse f⁻¹, if it exists (and is two-sided), is unique.
“While the notation f⁻¹(x) might be misunderstood, (f(x))⁻¹ certainly denotes the multiplicative inverse^[2] of f(x) and has nothing to do with the inverse function of f.”
“Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain” https://en.wikipedia.org/wiki/Inverse_function#Partial_inverses
“Left and right inverses are not necessarily the same.” https://en.wikipedia.org/wiki/Inverse_function#Left_and_right_inverses

Source: “Inverse Function,” Wikipedia, retrieved 11/7/2020, emphasis added.

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

“In mathematics, ‘symmetry’… is usually used to refer to an object^[1] that is invariant^[2] under some transformations…

“Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music…

“The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.”

This section of Wikipedia summarizes geometric symmetry quite well: https://en.wikipedia.org/wiki/Symmetry#In_geometry

“Generalizing from geometrical symmetry…, one can say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object.”

Source of information: “Symmetry,” Wikipedia, retrieved 11/7/2020, emphasis added.

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What is the Gist of an “Invariant” (Mathematics)?

“In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged, after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used.” For example, “the distance between two points on a number line is not changed by adding the same quantity to both numbers.”

“The phrases “invariant under” [a transformation] and “invariant to” a transformation are both used.” For example, the distance between two points on a number line is invariant under adding the same quantity to both numbers, but not under multiplying both numbers by the same quantity.”

Source: “Invariant (mathematics),” retrieved 11/7/2020, emphasis added

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