Tag: Functions

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:

“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.”

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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)?

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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 :).

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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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 “Injective,” “Surjective,” and “Bijective” Functions (Mathematics)?

This website sums it up well (and then goes into detail if you are interested):

“A function is a way of matching the members of a set “A” to a set “B”:

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Citation: Pierce, Rod. (19 Apr 2020). “Injective, Surjective and Bijective”. Math Is Fun. Retrieved 17 Oct 2020 from http://www.mathsisfun.com/sets/injective-surjective-bijective.html

Additional notes (from “Bijection,” Wikipedia, retrieved 10/17/20):

  • An injective function is also sometimes called “one to one” (because each B has at most one A and each A has exactly one B).
  • A surjective function is sometimes called “onto” (because every B has at least one A).
  • A bijective function is sometimes called a “bijection,” a “one-to-one correspondence,” or an “invertible 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. 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 :).