**Q1**. Which two of the following four regular expressions are equivalent? (ε is the empty string).

(i) (00)*(ε + 0)

(ii) (00)*

(iii) 0*

(iv) 0(00)*

(a) (i) & (ii) (b) (ii) & (iii) (c) (i) & (iii) (d) (iii) & (iv)

**(GATE-96)**

Sol:

Option (b) is correct.

Explanation:

i. (00)*( ε + 0) = (00)*ε + (00)*0 = (00)* + (00)*0 = 0*

Note that (00)* Generates strings of even length and (00)*0 generated the strings of odd length. Since + operator Union of both sets results in 0*.

**Q2**. The string 1101 does not belong the set represented by

(a) 110*(0+1)

(b) 1(0+1)*101

(c) (10)*(01)*(00+11)*

(d) (00+(11)*0)*

**(GATE-98)**

**Sol:**

Both option (c) and (d) will not generate the string 1101

**Q3**. if the regular set A is represented by A = (01+1)* an the regular set B is represented by B = ((01)*1*)*, which of the following is true?

(a) A Ì B

(b) B Ì A

(c) A and B are incomparable

(d) A = B

**(GATE-98)**

Sol:

Option d is true.

Explanation:

A = (01+1)*

A = ( (01)* 1* )* -- from rule (r1 + r2)* =( r1*r2*)*

A = B

Note:

If r

_{1}and r_{2}are two regular expressions then,
(r

_{1}+ r_{2})* = (r_{1}* + r_{2}*)*
= (r

_{1}* + r_{2})*
= (r

_{1}+ r_{2}*)*
= (r

(a) S Ì T

(b) T Ì S

(c) S = T

(d) S È T = f

Option c is true. From rule (r1 + r2)* =( r1 + r2*)*.

Note: Both S and T will generate the same language.

_{1}* . r_{2}*)***Q4**. Let S and T be languages over ∑ = {a,b} represented by the regular expression (a + b*)* and (a + b)*, respectively. Which of the following is true?(a) S Ì T

(b) T Ì S

(c) S = T

(d) S È T = f

**(GATE-00)****Sol:**

Option c is true. From rule (r1 + r2)* =( r1 + r2*)*.

Note: Both S and T will generate the same language.

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