I have multiple operands, say a, b, and c. I want an operator acting on a, b, and c, but the result should be invariant of order of operands (symmetric), and there should be no other pair with same result (unique).

What I tried:

Sum: a+b is symmetric, but 5+3 = 7+1. Similarly, product. I am unable to prove or disprove whether (a+b/ab) has the property I am asking for. Does anybody have any ideas about the function I am looking for?

  • $\begingroup$ What is the domain of your operator. $\endgroup$ – Don Thousand Mar 25 at 18:34
  • $\begingroup$ Integers basically numbers $\endgroup$ – Ajinkya Gawali Mar 25 at 18:52

If you want to define such a function for $a,b$ in the positive integers and with values in the positive integers, you can create it by filling in a table of values without repeating any values, except that the value for $(a,b)$ is the same as the one $(b,a)$. It might start out like this: $$ \begin{array}{c|ccccc} &1&2&3&4&5\\ \hline 1&1&2&4&7&11\\ 2&2&3&5&8&12\\ 3&4&5&6&9&13\\ 4&7&8&9&10&14\\ 5&11&12&13&14&15 \end{array} $$ In this example, the values in each row up to the diagonal are $1$; then $2,3$; then $4,5,6$, etc.

That defines a function which is symmetric, but otherwise does not repeat value.

If you want a formula, $$ f(a,b) = \begin{cases}\frac12 (a^2 - a) + b,& \text{ for } a \ge b\\ \\ \frac12 (b^2 - b) + a,& \text{ for } a \le b\end{cases} $$ or, without the cases, $$ f(a,b) = \frac14(a^2 + b^2 + a + b + (a + b - 3)|a-b|). $$


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