This post is about an interesting observation I made regarding rules and functions. One function can have two distinct rules. For example, consider the rules and . They are different, yet over the set , they define the same function. And the opposite can also happen. For example, the function over the reals is different than the function over the rationals , but they have the same rule, namely .
Of course, to make this all precise, we would need a rigorous definition of “rule” that can distinguish it from functions. I invite any readers to try to come up with a rigorous definition of “rule”.
Comments
Would you say that over the positive real numbers, the rule is the same as the rule ?
No, they are not the same rule, but they define the same function over the positive reals. Granted, this is a bit informal, because “rule” is not precisely defined, at least I have never seen a formal and rigorous definition of “rule”.