# Functions as sets

by cam875
Tags: functions, sets
 P: 230 Ive never been taught functions using sets and I have been told that to really understand the real meaning of them it is helpful to work them out and understand them with sets and stuff. Im not sure if im on the right track or confused but Im sure someone here can help me out. I have a basic understanding of sets and elements and all that so i should be able to follow along. thanks in advance.
 HW Helper P: 2,264 I am not sure how often the set definition of functions is helpful. There is also a notion of a function inducing a function from the power set of the domain into the power set of the range. It is helpful to first give the definition of a relation. Definition: Catesian Product Let A and B be sets. AxB is called the Cartesian product of A and B. AxB contains all ordered pairs of the form (a,b) were a and be are respectively elements of A and B. Definition: relation from A to B (set version) Let A and B be sets. R is a relation from A to B if R is a subset of AxB Definition: Domain (of a relation) Let R be a relation from A to B The domain of R, Dom(R) is the set of all elements a in A such that there exists a b in B so that (a,b) is in R. Definition: range (of a relation) Let R be a relation from A to B The range of R, Rng(R) is the set of all elements b in B such that there exists a in A so that (a,b) is in R. Definition: function from A to B (set version) let A and B be sets. A function f from A to B is a relation from A to B such that for each a in A there exist exactly one b in B such that (a,b) is in f. Often is is helpful to break this condition in two parts. Dom(f)=A (for each a in A there exist at least one b in B such that (a,b) is in f) if (a,b) and (a,b') are both in f, then b=b' (for each a in A there exist at most one b in B such that (a,b) is in f)
 P: 230 would you be able to give an example of a cartesian product between two example sets called A and B so that I could see the resulting set from that?