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In this paper, two reachability properties for single-input positive switched systems are introduced: monomial reachability and reachability. Monomial reachability, which represents a necessary but not sufficient condition for reachability, is fully characterized. Necessary and sufficient conditions for reachability are provided for the class of n-dimensional systems, switching among n subsystems. Several necessary or sufficient conditions are also provided. Moreover, the definition of k-switching reachability set is given, and an equivalent condition for the existence of an upper bound on the number of switchings required to reach any reachable vector is given. Finally, it is shown that, for reachable systems of low dimension (n = 2 or n = 3), each vector of the positive orthant can be reached by resorting to a switching sequence which switches no more than n - 1 times.