Tag: Operations

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)

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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 “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)

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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 “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 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 :).