Summary: Consider a continuous map \(f\colon X \to X\) and the set-valued map \(\bar{f}\) of \(\mathcal{K}(X)\) into itself induced by \(f\), where \(X\) is a metric space without isolated points and \(\mathcal{K}(X)\) is the space of all non-empty compact subsets of \(X\) endowed with the Hausdorff metric. In this paper, we discuss \(\mathcal{F}\)-transitivity and \(\mathcal{F}\)-mixing of the set-valued map \(\bar{f}\) induced by \(f\), where \(\mathcal{F}\) is a Furstenberg family. It is proved that, if \(\bar{f}\) is \(\mathcal{F}\)-transitive, then \(f\) is \(\mathcal{F}\)-transitive; moreover, \(\bar{f}\) is \(\mathcal{F}\)-mixing if and only if \(f\) is \(\mathcal{F}\)-mixing, and \(f\) is \(\mathcal{F}\)-mixing if and only if \(\bar{f}\) is \(\mathcal{F}\)-transitive, where \(\mathcal{F}\) is a full Furstenberg family.