The problem of assigning structural properties of a linear system through sensor selection is, for a given pair (A, B), to find an output pair (C, D) such that the resulting system (A, B, C, D) has the pre-specified structural properties, such as the finite and infinite zero structures and the invertibility properties. In this paper, by introducing the notion of infinite zero assignable sets for the pair (A, B), we establish necessary and sufficient conditions for the assignability of a given set of infinite zeros and a set of structural properties which includes the left invertibility property. In establishing these conditions, we develop a numerical algorithm for the construction of the required (C, D). (C) 2007 Elsevier Ltd. All rights reserved.