 \(\mathbf{C}^B\) is a choice function (i.e. satisfies the defining criteria for a choice function given in section 5.1) if and only if \(\succcurlyeq\) is complete and acyclical. It will then also satisfy properties \(\alpha\) and \(\gamma\). Furthermore, \(\mathbf{C}^B\) satisfies property \(\beta\) if and only if \(\succcurlyeq\) is transitive and complete (Sen 1970a, 19).