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The article establishes a constructive link between two approaches for analyzing the stability of discrete-time switched linear systems with arbitrary switching. The first approach is based on quasi-quadratic Lyapunov functions while the second one concerns a set-theoretic analysis of switched systems. An LMIs based procedure for deriving quasi-quadratic Lyapunov functions from the set-theoretic analysis is obtained. The approach may be used for constructing positively invariant sets for cases of switched discrete-time linear systems that are not quadratically stable.