Are there operations that can't be defined using a rule, and if they exist what is their significance? Wikipedia:Operations(mathematics)

In mathematics, an operation is a function which takes zero or more input values (called operands) to a well-defined output value

What I took away from this fact was that operations are essentially just functions.
I know that a function could just be an insignificant set of ordered pairs, with no formula associated with it, for example, $$\{(red,255),(green,127),(blue,0)\}$$
Are there functions that can't be described by anything more complex than sets, and if they do exist how would we define or use them? Would these "operations" be considered functions, or would we still consider them to be operations?
 A: In fact, the vast, vast majority of functions on, say $\mathbb R$, have no “rule”.
This is not very surprising when you realize that there are at least as many functions as there are real numbers (because there are constant functions spitting out any real number) and the vast, vast majority of real numbers have no “rule”. So, any given human description can be encoded in the Unicode character set which maps it to a unique integer in $\mathbb N$. But, it is known that the cardinality of $\mathbb R$ is simply a larger infinity than $\mathbb N$ as the binary expansion of a number in $[0, 1)$ (except for some details about repeating 1s at the end of an expansion, which turn out to be immaterial) can be viewed as specifying a subset of $\mathbb N$ consisting of all indices which have a 1-bit there. The set of subsets of $S$ is always a larger cardinality than $S$, they cannot be put into one-to-one correspondence; and so too with $S = \mathbb N$.
But, this is also true also for functions $\mathbb N \to \mathbb \{0, 1\}$ for example, and so the vast vast majority of functions from natural numbers to nontrivial sets are also indescribable in words or formulas.
Words and formulas can only exhaustively define all those functions from finite sets to sets which are either finite or countably infinite. Otherwise there are literally just not enough words to define some of the rules.
