Function equality
Shot in the dark, but maybe someone with good math theory lurks around here.
Had this discussion with a math tutor and need to double check.
We have two functions:
A- [1;3] -> [1;9] / x²
B- [1;3] -> R / x²
Tutor says these are different functions, I thought they were equal.
A≠B?
2 Replies
Different codomains. A is an onto function, B isn't.
What kind of notation is that anyway?
A : [1, 3] -> [1, 9], x -> x²
Formally a function is a set of ordered pairs of numbers subject to the restriction that if (x,y) and (x,z) are in the set, then y=z (formal way of saying that any given value can only appear once time as the first value of an ordered pair). We usually represent a function value for a given x, so formally a function can be defined as the set of ordered pairs (x,f(x)) for all values of x in the domain of the function.
By this definition of functions, the two functions you present consist of exactly the same set of ordered pairs. In this sense, then, the two functions are equal since set equality is defined such that sets A and B are equal if x is a member of A if and only if x is a member of B — that is both sets contain exactly the same elements.
The two functions you gave have different codomains but the same range. It would be reasonable to define functions f and g as equal if they have the same domain, the same range and f(x)=g(x) for all x in the domain. I wouldnÂ’t argue the point too hard because most of the time the formal definition of functions is ignored and functions such as the two you presented are often regarded as different.
The distinction is kind of in line with the distinction in set theory between intentional and extensional definitions of sets. It’s sort of a mathematical philosophy question, but modern math tends toward the extensional definition. Consider two sets. Set A is the set of all even prime numbers greater than 2. Set B is the set of all human beings who are capable of living unprotected on the bottom of the ocean without any life support equipment. Are sets A and B equal? The intensional position is that they are not — after all by their very definitions the represent different things. The extensional position is that both sets contain exactly the same elements, in this example no elements at all, and hence are equal.
Again, I wouldn’t belabor the point or argue with your tutor; this is getting into some pretty esoteric and mathematically formal stuff. But your intuition that the two functions is not unreasonable. It really just depends on how you define “equal” for functions.
