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Here the given language is $\phi$ which is empty set {} .

So , L={}=$\phi$

now , we know by definition for any language $L^{0}=\left \{ \epsilon \right \}$

$L^{1}=\left \{  \right \}=\phi$

$L^{2}=L.L=\left \{  \right \}\left \{  \right \}=\left \{  \right \}=\phi$

$L^{3}=L.L.L=\left \{  \right \}\left \{  \right \}\left \{  \right \}=\left \{  \right \}=\phi$

So ,

$\phi^{*}=L^{*}= L^{0}  \cup L^{1} \cup L^{2} \cup L^{3} \cup L^{4} \cup L^{5}\cup………….  $

 $\phi^{*}= \left \{ \epsilon \right \} \cup \left \{  \right \}\cup\left \{  \right \}\cup\left \{  \right \}\cup………...$

$\phi^{*}=\left \{ \epsilon \right \}$ 


$\phi^{+}=L^{1} \cup L^{2} \cup L^{3} \cup L^{4} \cup L^{5}\cup………….  $

 $\phi^{+}$ =$\left \{  \right \}\cup\left \{  \right \}\cup\left \{  \right \}\cup………...$

 $\phi^{+}$=$\left \{  \right \}$


For 3rd  I think the question wrong as $\phi$ is a set and $\epsilon$ is a string . So how can we define an Cartesian product of a set an a string ? Moreover Cartesian product term is wrong for language concatenation as Cartesian product will give us ordered pair not a language.

More details explained in the Link.

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∅^*:   having power ‘*’ will represent as {ε, s1,s2…..} i.e. ε will always present 

 therefore, for ∅^*: set will be { ε, ∅ } i.e. { ε , {} } => {ε}

 

∅^+:  having power ‘+’ will represent as ∅.∅^*

i.e. ∅.{ε,{}} but anything multiplied with ∅ will always gives ∅ i.e {empty} 

 

∅ is empty set {}

any Set multiply with ∅ will always give ∅ or {}

 

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