Systematically altering functions Let A and B be two integers, where their size is constrained to 2 bits in binary. There exists a function F, which outputs integers of the same size, where F(A) = F(B) = Y. For example, A = 01, B = 10, and Y = 11.

There exist some possible functions, such as G(X) = X xor 10. Another could be H(X) = X + 01 (ignoring overflow). However, in terms of their results, many of these would essentially be the same function, mapping the same inputs to the same outputs.

What is the "common denominator", so to speak, that would allow seeing all functions as the same?

The purpose of this is to be able to "increment" a function. In the first example, suppose I do not know what F is, and I want to write a program to find it. I know that F(A)=F(B), so I could write a loop to try all the possible functions by "incrementing" the function after each iteration, until the function that has F(A)=F(B) is found. Ultimately this will be used on larger sized, and possibly differently sized, inputs and outputs
You can represent a function by its truth table.  Then, any two functions with the same truth table will be the same function, in the sense that they map the same inputs to the same outputs, so this gives you a way to recognize those.

The truth table has 4 entries, each of which are 2 bits, so the truth table is 8 bits long.  Thus, each function can be associated with an 8-bit binary string, or equivalently, an integer between 0 and 255.  You can "increment" the function by incrementing the corresponding integer and converting it back to a truth table.