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The most elementary math operation always has two operands and an operator. For instance, the addition of two operands $a$ and $b$ can be represented as:

$$a + b, \quad a, b \in \mathbb{C}$$

This notation allows for the selection of any complex number $a$ and $b$ to perform addition but there is no flexibility in selection of the operator. Assume there is a set $\Omega$ that encompasses all possible operators ($+$, $-$, $\times$, $\div$, ^, √, %, $\dots$) so that we can achieve the form:

$$a\, O\, b, \quad O \in \Omega \text{ where } \Omega \text{ is the set of all possible arithmetic operators}$$

The idea behind using this form is that we can then define a set of infinite results obtained by performing a different combination of operator and second operand on a given number $x \in \mathbb{C}$ like so:

$$R_x = \{x \, O \, y \,|\, O \in \Omega, y \in \mathbb{C}\}$$

Can such a set $\Omega$ be precisely defined so that it encompasses every arithmetic operator that can be used between two operands? If this is possible, how can $\Omega$ be defined? If it cannot be defined, what are the limitations?

Matthew Towers
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Aniket Shinde
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  • You can just say "$\Omega$ is the set of all possible operators". What do you mean by "to define"? And why would such a set be useful? – HappyDay Oct 15 '23 at 10:28
  • It also seems that you are talking about operators that are closed in $\mathbb{C}$, in which case your $R_x$ is just $\mathbb{C}$... – HappyDay Oct 15 '23 at 10:32
  • By defining a set of operators, I mean to have a set such that we can take an element at a time to perform a unique operation between the operands that it operates on. So, +, −, ×, ÷, etc. If a rule can be written by which we can encompass all possible operations between two operands that would be the definition for $\Omega$. – Aniket Shinde Oct 15 '23 at 11:02
  • To answer why it is useful if $R_x$ is just $\mathbb{C}$, I want to add that each operation is mapped to its result in $R_x$ which in programming can be used to establish connections between seemingly unrelated data. If there exists can operation from which we can get to a given datapoint2 from datapoint1, we have established a meaningful relation between the two. I hope that kind of makes sense. This theory is still under development. – Aniket Shinde Oct 15 '23 at 11:09
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    Basically, you're asking for the set of all $2$ variable functions in $\mathbb{C}$. I don't think I understand what you mean by "rule". Why is "the set of all operations" not a good "rule"? – HappyDay Oct 15 '23 at 15:01
  • I want to define a set of operators precisely the same way we can define a set of natural numbers, for example. The term "set of all operators" is not a good definition because I cannot translate that into code. If I want to obtain an $R_x$ for a given $x$ using code, I would need to have a clearly defined set of operators. – Aniket Shinde Oct 16 '23 at 01:23

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