## UGC NET 2016 (July)

Question: The symmetric differences of two sets S1 and S2 are defined as:

S1⊕ S2 ={x ∣ x ∈ S1 or x ∈ S2, but x is not in both S1 and S2}

The nor of two languages is defined as

nor (L1, L2)={w ∣ w ∉ L1 and w ∉ L2}

Which of the following is correct?

A.  The family of regular languages is closed under symmetric difference but not closed under nor.

B.  The family of regular languages is closed under nor but not closed under symmetric difference.

C.  The family of regular languages are closed under both symmetric difference and nor.

D.  The family of regular languages are not closed under both symmetric difference and nor.

As we know Regular expression is closed under a symmetric difference () and nor (For details see Closer property of Regular Language).

Option C is correct.

Question: Consider the following two languages:

and ${\mathbf{L}}_{\mathbf{2}}$ is any subset of 0*.

Which of the following is correct?

(A)  L1 is regular and L2* is not regular

(B)  L1 is not regular and L2* is regular

(C)  Both L1 and L2* are regular languages

(D)  Both L1 and L2* are not regular languages

GCD of two infinite numbers is always Context - Sensitive Language (CSL) because multiplication and division of infinite number is not possible by regular language.

As we know that finite automata has finite memory that’s why it can’t perform infinite multiplication and division.

So,  is not regular.

A subset of a single symbol of kleene star operation 0* is always regular language.

So, L2 is any subset of 0* is regular language.

Option (B) is correct.