Simple Linear Algebra Problem... is the book wrong?

Let V denote the set of ordered pairs of real numbers. If (a, b) and (c, d) are elements of V and x is an element of R, define:

(a, b) + (c, d) = (a+c, bd)

and

x(a, b) = (xa, b).

Is V a vector space over R? Justify.

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Well, it isn't, but the book claims this is due to the fact (which is referred to by (VS4)), that for each element a in V there exists an element b in V such that

a + b = 0.

However, if c = -a, and d = 0, then (a+c, bd) = (0, 0) always. Am I missing something here?