If g is a function that maps some elements of a set A to elements of a set B, then the domain of g is the subset of A for which g(a) is defined.
More formally:
g sube A xx B :
AA a in A AA b_1, b_2 in B
((a, b_1) in g ^^ (a, b_2) in g) => b_1 = b_2
Use the notation 2^A to represent the set of subsets of A and 2^B the set of subsets of B.
Then we can define the pre-image function:
bar(g)^(-1): 2^B -> 2^A by bar(g)^(-1)(B_1) = {a in A : g(a) in B_1}
Then the domain of g is simply bar(g)^(-1)(B)
If f is a function that maps some elements of set B to elements of a set C, then:
bar(f)^(-1): 2^C -> 2^B is defined by bar(f)^(-1)(C_1) = {b in B : f(b) in C_1}
Using this notation, the domain of f@g is simply
bar(g)^(-1)(bar(f)^(-1)(C)) = (bar(g)^(-1)@bar(f)^(-1))(C)
