$\text{Why DCFL is closed under complemetation?}$
$\text{we can easily find its complement by interchanging Non final to final and vice versa}$
$\text{then why not CFL is closed under complementation?}$
$\text{because interchanging Non final -final and vice versa will not gurantee to give the complement}$
$\text{as it is non -deterministic }$
$\text{Why DCFL is NOT closed under intersection?}$
$\text{i will need just 1 counter example to prove that is not closed as closure property says that }$
$\text{this property should be maintained for every example}$
$L_{1}=a^{n}b^{n}c^{m}\text{(DCFL)}$
$L_{2}=a^{m}b^{n}c^{n}\text{(DCFL)}$
$L_{3}=a^{m}b^{n}c^{m}\text{(DCFL)}$
$L_{counter}=L_{1} \cap L_{2} \cap L_{3} = a^{s}b^{s}c^{s}$
$L_{counter} \text{is not DCFL/CFL}$
$\text{Why DCFL is NOT closed under Union?}$
I will prove it using Proof by contradiction and will use above $2$ results
$\text{Let} L_{union} \text{is DCFL}.$
$L_{union} =L_{1} \cup L_{2} \cup L_{3}$
$L_{union}^{'} =L_{1}^{'} \cap L_{2}^{'} \cap L_{3}^{'}$
$\text{let} L_{union}^{'} =L_{Ucomp} \text{which will be DCFL as it is closed under complement}$
$L_{Ucomp}=L_{comp1} \cap L_{comp2} \cap L_{comp3}$
$\text{where} L_{comp1} , L_{comp2}, L_{comp3} \text{are DCFL}$
$\text{hence it proves that DCFL are closed under intersection ,contradiction)}$