Understanding the Function of Set S in Discrete Mathematics

[Nert]

Hey guys,

I was reading Kenneth's Discrete Mathematics and I came across this definition in the function chapter:

Let f be a function from A to B and let S be a subset of A.The image of S under the function f is the subset of B that consists of the images of the elements of S.We denote the image of S by f(S), so f(S) = {t | ∃s∈S (t = f(s))}.
We also use the shorthand {f(s) | s ∈ S} to denote this set.

My questions is:
1) What is the purpose of set S?

From my understanding, S is just a subset of A which has corresponding image for each element of S?

 

[Simon Bridge]

1) What is the purpose of set S?

To provide for an object that the definition can be applied to.
As written it has no purpose outside of the definition.

From my understanding, S is just a subset of A which has corresponding image for each element of S?

... the definition is for what is meant by the image of S. The image of S under f is T, which is the subset of B with elements t=f(s).

 

[Fredrik]

The definition defines f(S) for all S such that S⊆A. Compare this to how you can define a function g by g(x)=x² for all real numbers x. What is the purpose of the real number x? That would be a strange question, since we haven't defined a number x. We have only defined a function g.

 

[PeroK]

What is the purpose of f? Or the sets A and B?

 

[Stephen Tashi]

What is the purpose of f? Or the sets A and B?

In more advanced mathematics there are lots of questions that deal with whether certain kinds of functions preserve certain kinds of structures. For example, does a linear tranformation map a vector space to a vector space? The structures are usually some kind of set or subset that has special properties. Hence it's convenient to have terminology for the image of a set by a function.