I) Let E be a correspondence from U to X. If (s_{0}, x_{0}) ∈ (ED)C then there is a t_{0} ∈ T such that (t_{0}, x_{0}) ∈ ED and (s_{0}, t_{0}) ∈ C. And there is a u_{0} ∈ U such that (u_{0}, x_{0}) ∈ E and (t_{0}, u_{0}) ∈ D. So (s_{0}, u_{0}) ∈ DC and (s_{0}, x_{0}) ∈ E(DC). Thus (ED)C ⊂ E(DC). If (s_{1}, x_{1}) ∈ E(DC) then there is a u_{1} ∈ U such that (u_{1}, x_{1}) ∈ E and (s_{1}, u_{1}) ∈ DC. And there is a t_{1} ∈ T such that (t_{1}, u_{1}) ∈ D and (s_{1}, t_{1}) ∈ C. So (t_{1}, x_{1}) ∈ ED and (s_{1}, x_{1}) ∈ (ED)C. Thus E(DC) ⊂ (ED)C. Therefore (ED)C = E(DC).

II) *To do*

Advertisements