Question

9. Which of the following are true for all regular languages and all homomorphisms? (a) h (L1 ∪ L2)= h (L1) ∩ h (L2). (b) h (L1 ∩ L2)= h (L1) ∩ h (L2). (c) h (L1L2) = h (L1) h (L2).

Answer 1

Answer:

h (L1 ∪ L2)= h (L1) ∩ h (L2).

This is true.

There will be w ∈  (L1 ∪ L2) for any s ∈ h (L1 ∪ L2) in such a way that s=h(w)

we can assume that w ∈ L1

So In this case h(w) ∈ L (S1). Hence s ∈ L(S1)

for any s ∈ h (L1) U h(L2)

We can assume that s ∈ L(S1)

There exists w ∈ L1 such that s= h(w)

In this case it is w ∈ L1 U L2 as well.

Hence , s ∈ h (L1 U L2)

Explanation

consider  = 0,1 and  = a,b and h(0) = a , h(1) = ab

(a) Consider L1 = 10,01 and L2 = 00,11

   Now L1 ∪ L2 = 00,01,10,11

   h (L1 ∪ L2) = h(00) , h(01) , h(10) , h(11) = h(0)h(0) , h(0)h(1) , h(1)h(0) , h(1)h(1)

  = aa, aab , aba , abab

Hence h (L1 ∪ L2) = aa, aab , aba , abab .

Here h (L1) = h(10) , h(01) = h(1)h(0) , h(0)h(1) = aba , aab

Hence h (L1) = aba , aab .

Here h (L2) = h(00) , h(11) = h(0)h(0), h(1)h(1) = aa, abab

Hence h(L2) = aa, abab.

Finally Hence , h (L1 ∪ L2)= h (L1) ∩ h (L2).

Answer 2

The option that is true about regular languages and all homomorphisms is h(L1L2)=h(L1)h(L2).

What is homomorphism?

The term homomorphism is known to be an algebra term that connote a form of  a structure-keeping map that exist between two algebraic structures of the same kinds (such as two groups, two rings, etc.).

Note that the word homomorphism originates from the Ancient Greek language called : ὁμός (homos) that implies the "same" and μορφή (morphe) that implies "form" or can simply say  "shape".

Conclusively, The option that tells more about regular languages and all homomorphisms is h(L1L2)=h(L1)h(L2).

Learn more about homomorphisms from

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