Inverse Homomorphism expression

Consider the following expression :- $$h(h^{−1}(L))$$

I need an example where this expression can be superset of subset of L,but i am not able to get one.I am getting this equal to L always.How can i approach this?

• It is always $h(h^{-1}(L)) \subseteq L$ and $h^{-1}(h(L)) \supseteq L$. – fade2black Aug 3 '17 at 23:45
• I was looking for some example and i got one as:- Let h(0) = ab; h(1) = ε. Let L = {abab, baba},h-1(L) = the language with two 0’s and any number of 1’s = L(1*01*01*). But i didn't understand here is that how can baba be there in L as L must have been obtained by applying homomorphism to some language L' ,but non of homomorphic functions matching to baba.Can you please explain this? – rahul sharma Aug 4 '17 at 1:26
• $L$ is not obtained by the homomorphism. $L$ is given as well as the homomorphism. When you apply $h(h^{-1})$ to $L$ you obtain a proper subset of $L$, $\{abab\}$. – fade2black Aug 4 '17 at 2:02
• If h(0)=ab then can i have any language which contains ab?Or does it have to be some valid language that is obtained after applying h(L) ? – rahul sharma Aug 4 '17 at 9:56
• I don't understand what you want to solve. Could you please rephrase your problem by editing your OP? – fade2black Aug 4 '17 at 10:13