each of the three sets given are defined by properties. for example:

1. S = {f in F[0,1] : f(0)f(1) = 0}.

this means that S is the set of all real-valued functions defined on the interval [0,1], for which either f(0) or f(1) (or both) is 0.

the 0-vector of F[0,1] is the function z, where z(x) = 0 for all x in [0,1]. clearly this function is 0 at x = 0 and x = 1.

now suppose f(x) = x. since f(0) = 0, f(0)f(1) = 0, so f is in S. suppose further, that g(x) = 1-x. then g(0)g(1) = (1)(0) = 0, so g is in S.

but (f+g)(x) = x + 1 - x = 1, so (f+g)(0)(f+g)(1) = (1)(1) = 1.