Prove that in a given vector space V, the zero vector is unique.

Suppose, by way of contradiction, that there are two distinct additive identities 0 and u₀. Then, which of the following is true about the vectors 0 and u₀? (Select all that apply.)

The vector u₀ is equal to 0.The vector 0 + u₀ is not equal to u₀ + 0.The vector 0 + u₀ is not equal to u₀.The vector 0 + u₀ is equal to u₀.The vector 0 + u₀ is equal to 0.The vector 0 + u₀ is not equal to 0.The vector 0 + u₀ does not exist in the vector space V.

This contradicts which of the following assumptions? (Select all that apply.)

The vectors 0 and u₀ are additive identities.The vectors 0 and u₀ are distinct.The vectors 0 and u₀ are additive inverses.The vectors 0 and u₀ are vectors in the vector space V.The vectors 0 and u₀ are multiplicative identities.

Therefore, the additive identity in a vector space is unique.