Tag: Binary 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 :).