Author: George Evans

BS in Physics with a Minor in Mathematics.

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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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 a “Group” (Group Theory)?

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

More precisely, “a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms[3] are satisfied, namely closure,[4] associativity,[5] identity[6] and invertibility.[7]” (“Group (mathematics),” Wikipedia, retrieved 10/17/2020)

I particularly like Wikipedia‘s example of groups and their definition of groups that follows, so I’ve included the information below:

“Example: the integers

“One of the most familiar groups is the set of integers \mathbb {Z}  which consists of the numbers

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …,

 together with [the binary operation] addition.

“The following properties of integer addition serve as a model for the group axioms given in the definition below.

  • “For any two integers a and b, the sum a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition.
  • “For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.
  • “If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.
  • “For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

“The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

Definition

“A group is a set, G, together with [a binary operation “⋅”]… that combines any two elements a and b to form another element, denoted a ⋅ b or ab. [Note that ⋅ is called the group law of G]. To qualify as a group, the set and operation, (G, ⋅), must satisfy four requirements known as the group axioms:

  • “Closure: For all a, b in G, the result of the operation, a ⋅ b, is also in G.
  • “Associativity: For all a, b and c in G, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).
  • “Identity element: There exists an element e in G such that, for every element a in G, the equation e ⋅ a = a ⋅ e = a holds. Such an element is unique… and thus one speaks of the identity element.
  • “Inverse element: For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted “+”), such that a ⋅ b = b ⋅ a = e, where e is the identity element.

…

“The set G is called the underlying set of the group (G, ⋅). Often the group’s underlying set G is used as a short name for the group (G, ⋅). Along the same lines, shorthand expressions such as “a subset of the group G” or “an element of group G” are used when what is actually meant is “a subset of the underlying set G of the group (G, ⋅)” or “an element of the underlying set G of the group (G, ⋅)”. Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.”

Source: “Group (Mathematics),” Wikipedia, retrieved 10/17/2020

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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 :).

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

Very loosely speaking, the inverse element undoes the effect of combining two elements, and elements combined with their inverse element return the identity element.[1]

To be more precise, suppose there exists a binary operation[2] acting on a set[3]. If an element of this set is put into the binary operation with the element’s inverse, the result is the identity element of the set. Furthermore, if an element is combined with another element of the set via a binary operation, combining the result with the original element’s inverse will return the second element, e.g., 5+6=11, then 11+(-5)=6, where 5 was the original element, (-5) is the inverse of 5 under the addition operation, and 11 was the result of the original combination.

If an element has an inverse, it is considered “invertible.” If every element in a set has an inverse, the set is considered to have the property of invertibility.

Additional points to note:

  • In the above definition, we assumed the set is “closed”[4] under the binary operation and that there existed a two-sided identity element in the set.
  • Generally, no two elements in a set have the same inverse element.
  • If the order that an element goes into the binary operation matters, there can be elements which behave like an inverse element if put in first but not when put in second. It’s also possible to have elements which behave like an inverse element if put in second but not when put in first.
  • The inverse element of arbitrary element a is frequently called a-1.
  • For examples of inverse elements, see https://en.wikipedia.org/wiki/Inverse_element#Examples.
  • It’s also possible to have more generalized version of an inverse without an identity. For more information, see this link.

Source: “Inverse Element,” Wikipedia, retrieved 10/25/2020

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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 :).

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 :).

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

“An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments….

“As used in mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”.” Logical axioms seems to be about things that are always true, such as tautologies. Non-logical axioms seem to be about things which are true in certain situations, such as a + b = b + a. (“Axiom,” Wikipedia, retrieved 10/17/20, emphasis added)

When referring to non-logical axioms, the words “axiom”, “postulate”, and “assumption” are interchangeable. (Ibid)

“In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow — by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.”

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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 “Identity” (Mathematics)?

When one of the two inputs of a binary operation[1] that is acting on a set[2] is an identity element, the output will be the other element that was used as an input.

Very loosely speaking, the identity element doesn’t change anything when combined with things. E.g., multiplying 3 by 1 (a multiplicative identity) results in 3, or adding 0 (an additive identity) to 5 results in 5.

Additional points to note:

  • Which element of the set is the identity element depends on which binary operation is acting on the set.
  • The identity element is often just called “the identity” if it is clear which binary operation and set you are talking about.
  • If the order that an element goes into the binary operation matters, there can be elements which behave like an identity element if put in first but not when put in second. It’s also possible to have elements which behave like an identity element if put in second but not when put in first, e.g., 5-0=5 but 0-5=-5.
  • It is only possible for there to be zero or one identity elements which behave like an identity element regardless of the order it is put into the binary operation.
  • For examples of identity elements, see https://en.wikipedia.org/wiki/Identity_element#Examples

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Source: “Identity element,” Wikipedia, retrieved 10/10/2020

[1] See “What is the Gist of a “Binary Operation”?“

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

Some binary operations[1] have something called “the associative property… which means that rearranging the parentheses in an expression will not change the result.”

To elaborate, let “*” be a binary operation[1]. The operation “*” is associative when acting on a specific set[2] S only if the following statement is always true: (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z in S

Note: associativity means something different in propositional logic, and associativity is not the same thing as commutativity[3].

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Source of quotes and information: “Associative property,” Wikipedia, retrieved 10/10/2020. Bold added to quote for emphasis.

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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 “Closure” (Mathematics)?

“A set[1] is closed under an operation if performance of that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction: 1-2 is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because 0+0=0, 0-0=0, and 0\times {0}=0).

“Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually” (“Closure (mathematics),” Wikipedia, retrieved 10/9/2020, bold added or taken away for emphasis).

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

A binary operation takes two things as an input (called operands) and produces another thing as a final result (“Binary operation,” Wikipedia).

If a you perform a binary operation on a set[1], then (usually) both operands and the final product are in that same set (ibid.). To elaborate, the “two domains and the codomain [of the operation] are [usually] the same [as the set being operated on]. Examples include the familiar arithmetic operations of addition, subtraction, multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.” (ibid.)

All quotes were retrieved 10/8/2020.

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

“In mathematics, a set is a well-defined collection of distinct* objects**, considered as an object** in its own right….

“…The objects that make up a set (also known as the set’s elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on…

“…Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.”

Source: “Set (Mathematics),” Wikipedia, retrieved 10/3/2020, bold added or taken away for emphasis.

As of 10/3/2020, https://en.wikipedia.org/wiki/Set_(mathematics)#Set_notation has a nice summary of the notation used to define a set.

* The fact that the objects** are distinct means that none of the objects in the set are equal to each other.

**As for what a mathematical object is, see What is the Gist of a “Mathematical Object”?

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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 :).