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Q: Does the set of rational numbers have an additive inverse?

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Yes.

Wrong! Not only is zero a real number, but it is the additive identity for the set of integers, rational numbers as well as real numbers.

Yes, it does.

Yes, it does.

There is only one set and it does have an additive identity.

Yes, and for any non-zero rational x, the multiplicative inverse is 1/x.

It gives closure to the set of real numbers with regard to the binary operation of addition. This makes the set a ring. The additive inverse is used, sometimes implicitly, in subtraction.

Yes. The additive identity is 0.

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Inverse operations. Additive inverse is not one operation but they are elements of a set.

I t is the number 0, which has the property that x + 0 = 0 + x = x for all rational numbers x.

Yes it has closure, identity, inverse, and an associative property.

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