Axiom number 4 for vector space states that "there exists a zero vector 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u, for all u in V.
My questions is, must all the elements in the zero vector be zeroes?
For this question: "Given the set of all pairs of real numbers of the form (x, 1) with the operations (x, 1) + (x', 1) = (x + x', 1)", what is the zero vector for this case, if it exists? Is it (0, 0) or (0, 1)? Also, is it necessary for the zero vector to be contained in V?
I think that the zero vector exists, which is (0, 1), and all zero vectors for any vector spaces must be contained in V. But then again, the definition for zero vector is that all the elements / components must be zeroes.