Let take L = { W1 , W2, W3, W4, ........}
my convention is ( Wi )R = Xi.
we know that (W1.W2 )R = (X2 . X1)
ex:- let W1 = ab ( this means X1 = ba ), W2 = cd ( this means X2 = dc ) ===> (W1 . W2)R = (ab.cd)R = (dc . ba) = (X2. X1)
what is L* ?
L* ={ ∈, L , L2, L3,.......} (simply concentrate on L2 only )
= { ∈, ( W1 , W2, W3, W4, .......) , ( W1W1 , W1W2, W1W3, W1W4, .......) , ( W2W1 , W2W2, W2W3, W2W4, .......) , ..... }
( L* )R = simply reverse each string in the L*
= { ∈, ( (W1)R, (W2)R, (W3)R, .......) , ( ( W1W1)R , (W1W2)R, (W1W3)R, .......), ( ( W2W1)R , (W2W2)R, (W2W3)R, .......), ..... }
= { ∈, ( (X1), (X2), (X3), (X4), ...........) , ( ( X1X1) , (X2X1), (X3X1), .................), ( (X1X2) , (X2X2), (X3X2), , .......), ..... }
what is LR ?
LR = reverse each element in the language
= { (W1)R, (W2)R, (W3)R, ......}
= { (X1), (X2), (X3), (X4), ........... }
(LR)* = {∈, ( (X1), (X2), (X3), (X4), ...........) , ( ( X1X1) , (X1X2), (X1X3), .................), ( (X2X1) , (X2X2), (X2X3), , .......), .....}
= { ∈, ( (X1), (X2), (X3), (X4), ...........) , ( ( X1X1) , (X2X1), (X3X1), .................), ( (X1X2) , (X2X2), (X3X2), , .......), ..... }
= (L*)R
NOTE :-
word Xi . Xj is in ( LR )* it must be in ( (L*)R) converse also.
think why this is true? if you didn't get in general, take the sample strings and realize how it should be true.