Math 323 Comments on Codomains and Domains 21st Century

See the definition of function at the link Functions, Definitions on the class website, quoted here: Given sets A and B, a relation F is said to be a function from A to B [notation: F : A ! B] iff • F is a relation between A and B; • F has the function property; • dom (F) = A; i.e., for all x in A, there is y in B such that (x, y) " F ( see definition of domain [at notes cited above]). (The last part of the definition is not part of the definition in the textbook; in the textbook, it is simply required that dom(F) # A . ) What does this mean for the sets A (the domain) and B (the codomain)?

CODOMAIN: The role of the codomain B is different from the role of the domain A (see discussion of domain below). There are many sets which could be used as a codomain for a given set of ordered pairs F with the function property. By definition, the notation F : A ! B means (among other things) that F is a relation between A and B, this is the ONLY place B appears in the definition. So for every pair (x, y) in F, (x, y) " A$B; in other words, for all x in A, (x, F(x)) is in A$B. In the standard function notation, this means that for all x in A, F(x) is in B. This is the ONLY condition imposed on B by the notation F : A ! B. The most serious error students make with interpreting the notation, F : A ! B, is to think that this means that B is (must be) the range of F. This is a serious error. Regarding B, a necessary and sufficient condition for the statement F : A ! B to be correct is that stated above: For all x in A, F (x) is in B. (B can be much bigger than the range.) Of course, this is true if and only if ran(F) # B. So students often thank that they have to know the range and THEN prove ran(F) # B. NO. In order for B to be an acceptable codomain just show for all x in A, F(x) is in B .

DOMAIN (both paragraphs below are important): Given a relation F with the function property, then by the definition above, the notation F : A ! B simply means, for A, that A is the domain of F -- i.e., we have chosen to use A to denote the domain of F ; this is the way we use the notation F : A ! B in this class, and it is a common notation). The relation (function) F has a domain; the domain is an intrinsic part of the relation (function); A is not something we have added to the information given by F; it is simply a name for the domain of F. Of course, this means, in function notation, that, for each a in A, there must exist F(a) (the second element of the ordered pair (a, F(a)) in the relation). If this is not the case, there is something wrong with our notation, and our set A is not, in fact, the domain of F. To put this in other words, WE do not decide what the domain is; the function F decides; and then we have chosen to use A as the name of the set which is the domain. The discussion above applies when we are given a function (a relation with the function property) F. The point of view is somewhat different when a function is being constructed (or defined, or manufactured). When doing this, one often starts with the set, say A, which we want to be the domain, and then for each x in A, we give a “rule” for determining F(x). If we have done things correctly, then the set A we started with will be the domain. But then A is the domain of the function, and it “belongs to the function”. If we change the domain, we change the function.