Does adding two hash values generate a valid hash?

Question

Does adding 2 hash values generate another valid hash value? In other words will this hash(a) + hash(b) != hash(c) + hash(d) always be true? I don't think it will but does it matter? Are the essential properties of the hash function preserved under addition?

Answer 1

Since several values can have the same hash, it could be that hash(a) = hash(b) = hash(c) = hash(d), so also hash(a) + hash(b) = hash(c) + hash(d).