Difference between codomain and range

  • Thread starter Thread starter smithnya
  • Start date Start date
  • Tags Tags
    Difference Range
AI Thread Summary
The discussion clarifies the difference between codomain and range in mathematical functions. The codomain is the set of all possible outputs a function can produce, while the range consists of the actual outputs for a given domain. For the function f(x)=sin(x), the codomain is R, but the range is [-1,1] when the domain is all real numbers. If the domain is restricted to [0, π/2], the range changes to [0,1]. The distinction emphasizes that the range is dependent on the specific domain chosen for the function.
smithnya
Messages
41
Reaction score
0
I am having a difficult time wrapping my mind around the differences between a codomain and a range. Could someone explain the difference between the two and possibly provide an example?
 
Physics news on Phys.org
Consider the function f(x)=sin(x)

We could say f:R->R. The codomain of f is R because that's where the values of x are mapped to.

The range of f:X->Y, S, is defined as: for all elements y in Y, there exists x in X such that y=f(x). Basically, S=f(X).

In our example above, the range is [-1,1].
We have π mapped to 0, π/2 mapped to 1, etc.


Note that the range is dependent on the domain.

Edit: If you consider g:[0,π/2]->R, then the range of g(x)=sin(x) is [0,1].

If you consider h:R->C, the range of h(x)=sin(x) is still [-1,1]. The codomain being the complex numbers. Since the domain are reals, sin maps them to real values.
 
Last edited:
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
Back
Top