Let be a set of functions with a common domain and a common range . A set is called a set of range uniqueness (SRU) for , if for all , Let be the set of all real polynomials in variables of degree at most and let be the set of all linear functions with rank . We show that there are SRU's for of cardinality , but there are no such SRU's of size or less. Moreover, we show that there are SRU's for of size but there are no such SRU's of smaller size.