4.2.15: (a) First, Dom(f x g) = A x C. Indeed, let (a,c) be an element of A x C. Since Dom(f)=A, there exists b in B such that (a,b) is an element of f, and similarly there is a d such that (c,d) is an element of g. Then ((a,c),(b,d)) is an element of f x g. To see that f x g satisfies the vertical line test, suppose ((a,c),(b,d)) and ((a,c),(b',d')) are elements of f x g. Then (a,b) and (a,b') are elements of f, so b=b' since f is a function, and (c,d) and (c,d') are elements of g, so d=d' since g is a function. Then (b,d)=(b',d'). Thus f x g is a function.

(b) We have ((a,c),(b,d)) in f x g if and only if (a,b) in f and (c,d) in g, if and only if b=f(a) and d=g(c). Then (f x g)(a,c)=(f(a),g(c)).