UGC NET CSE | January 2017 | Part 3 | Question: 20

Question

Consider the languages $L_{1}= \phi$ and $L_{2}=\{1\}$. Which one of the following represents $L_{1}^{\ast}\cup L_{2}^{\ast} L_{1}^{\ast}$?

• A. $\{\in \}$
• B. $\{\in,1\}$
• C. $\phi$
• D. $1^{\ast}$

Comments

L ₁* ∪ L ₂* L ₁*

 φ* ∪ (1)*.φ*

* > . >  ∪ (priority)

ε ∪ (1)*.ε

ε ∪ (1)*

1*

Answer 1

option 4

L1 = ϕ, and L2 = {1}
L1 * is also ϕ
L2* will be 1*
so L1* U L2* L1* = 1*

Comments

L1* is not ϕ but {ε}

Answer 2

$L_1^* = \phi ^* = \epsilon$

$L_2^*L_1^* = \{1\}^* . \epsilon = 1^*$

$L_1^*\ \cup L_2^*L_1^* = \epsilon \ \cup 1^* = 1^*$

Option (D)

Answer 3

∅*=∈

∅.anything =∅

∈ union ∅ =∈

A is answer

Comments

As, L1*  = epsilon

  therefore, L2*. L1* = 1* . epsilon ( not 1*. phi )
         which results in 1*

     i. e Option D.  ( not option A)

Answer 4

4 is answer

because l1 is empty so no use of union and l2 is singleton so it will be 4