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Which of the following are regular sets?

1. $\left\{a^nb^{2m} \mid n \geq 0, m \geq 0 \right\}$

2. $\left\{a^nb^m \mid n =2m \right\}$

3. $\left\{a^nb^m \mid n \neq m \right\}$

4. $\left\{xcy \mid x, y, \in \left\{a, b\right\} ^* \right\}$

1. I and IV only
2. I and III only
3. I only
4. IV only

Why iv is regular anyone explain
Cz u can have a regular expression for iv

(a+b)*c(a+b)*

But by pumping lemma it can be shown that a^n b^2m is not regular. Please correct me if I'm wrong.

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Since in option $2$ and $3, n$ is dependent on $m$, therefore a comparison has to be done to evaluate those and hence are not regular.

I and IV are clearly regular sets.

if option D were
xcx then both x,c should belong to (a+b)* , right ?
how option d is regular here ? for this c should belong to (a + b)^(*) it is not as it will give only c on keeping x as epsilion
@Leensharma can you explain option d)
d) *IS* regular. The regular language for d) will be (a+b)*c(a+b)*
can you explain how 1st is regular by some explanation?

@Shubham Aggarwal I Think first can be written as -->(a)*(bb)* . Here both asterisks are independent of each other and can be thought as n and m.

But by pumping lemma it can be shown that a^n b^2m is not regular. Please correct me if I'm wrong.

I)Set of all strings containing any number of ‘a’ s followed by an even number of ‘b’ s. R.E=(a)$^{*}$(bb)$^{*}$.

IV) Strings containing a ‘c’. R.E= (a+b)$^{*}$c(a+b)$^{*}$.

Both these languages are regular as regular expressions exist.

By default a language is infinite. Eg : {a$^{n}$} it’s a infinite language.So both the languages II and III are infinite and comparison has to be done to evaluate these and hence are not regular.

NOTE:

Every finite language is regular.

Infinite language + Comparison = Non-Regular.

Infinite language + No Comparison = Regular.

Edit: As nothing is mentioned about ‘c’ in option IV and there is a comma after y, So I think It’s a typo ‘c’ should also belongs to {a,b}$^{*}$. IV will be a complete language. Which is regular. R.E=(a+b)$^{*}$.