6. The symmetric difference of two sets S 1 and S 2 is defined as
S 1 θ S 2 = {x: x ∈ S 1 or x ∈ S 2 , but x is not in both S 1 and S 2 }.
Show that the family of regular languages is closed under symmetric
difference.
7. The nor of two languages is
Show that the family of regular languages is closed under the nor operation.
8. Define the complementary or (cor) of two languages by
Show that the family of regular languages is closed under the cor operation.
9. Which of the following are true for all regular languages and all
homomorphisms?
(a) h (L 1 ∪ L 2 )= h (L 1 ) ∩ h (L 2 ).
(b) h (L 1 ∩ L 2 )= h (L 1 ) ∩ h (L 2 ).
(c) h (L 1 L 2 ) = h (L 1 ) h (L 2 ).
10. Let L 1 = L (a*baa*) and L 2 = L (aba*). Find L 1 /L 2 .
11. Show that L 1 = L 1 L 2 /L 2 is not true for all languages L 1 and L 2 .
*12. Suppose we know that L 1 ∪ L 2 is regular and that L 1 is finite. Can we
conclude from this that L 2 is regular?
13. If L is a regular language, prove that L 1 = {uv : u ∈ L, |υ| = 2} is also
regular.
14. If L is a regular language, prove that the language {uv : u ∈ L,υ ∈ L R } is
also regular.
15. The left quotient of a language L 1 with respect to L 2 is defined as
Show that the family of regular languages is closed under the left quotient