The authors investigate the smallest number \(s_t\) of blocks identifying a not necessarily simple \(t\)-\((v,k,\lambda_t)\) design \({\mathcal D}\). Such a multiset of blocks identifying \({\mathcal D}\) is called a defining set. The authors use trades, i.e. a pair of families of \(k\)-subsets both covering each \(t\)-subset equally often, to derive bounds for \(s_t\). An interesting bound is \(s_s\geq s_t+ 2^{t-s}-1\), where \(s_s\) is the number of blocks in a smallest defining set of \({\mathcal D}\) when \({\mathcal D}\) is considered as an \(s\)-design for \(0<s<t\). Further bounds concern the block fraction needed for a smallest defining set for \(v\leq 8\) and for Steiner designs.