homomorphism

Question

Consider the homomorphism, h(0) = aba, h(1) = abb
Given L = (a + bb)^*ba, which of the following represent the h^-1(L)

1.  0*
2.   100*
3.   1*0
4.   11*0

Comments

Any resource about that.

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inverse homomorphism need not contain mappings for all the strings of L, some strings might be lost during inverse homomorphism case.

this is main reason why h(h⁻¹(L)) can be a subset of L  is true, otherwise, it would be,

h(h⁻¹(L)) = L..

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You're right nitish actually.

in general h (h^-1(L)) ⊆ L ⊆ h^-1(h(L) )

h^-1(L)={w∈ Σ^* | h(w)∈ L}

option a) h^-1(L)=0^*  it contains string 00 and h(00)=abaaba which doesn't belong to our L

option b ) h^-1(L)=100^* it contains string 100 and h(100)=abbabaaba which doesn't belong to our L

option d) h^-1(L)=11*0 it's subset of option c , so see if string that doesn't belong to option d is also supported then , answer is option c. 0 is string that belongs to option c but not d and it's supported

so answer is option c , albeit i don't know way to directly tell h^-1(L)

i'm attaching source to understand homomorphism , those problems are quite less important from GATE point of view , so it's better to don't put much research here.

https://courses.engr.illinois.edu/cs373/sp2013/Lectures/lec08.pdf