Made easy test series

Question

Comments

Yes, it is like comparison but  $n$ and $m$ are non-negative integers.
So  $n = m = 0$ and $n= 1 , m=k , k \geq =1 $ covers all the strings of type $a^k , k \geq 0.$

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Okk so in these type of question we have to interpret the question by considering all the possibility?

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Yes, We have to consider all possibilities. That's the trick here.
A simple way to understand such type of questions is start enumerating the strings.
like $S_1 = \{ a, aa,aaa,aaaa...........\}$ . You can easily find that strings of every length $\geq 1$ can be obtained here .

Answer 1

Both of the them are regular

Here S2 = {aⁿbⁿ | n>=1} ∪ { aⁿb^m | n>=1,m>=1 } = { aⁿb^m | n>=1,m>=1 } which is regular.

And S1 = { (aⁿ)^m | n <= m >= 0 } 

Let aⁿ = b then S1 becomes 

S1 = { b^m | n <= m >= 0 } which is also regular.

Look at the image below

Here FIG A gives the DFA for S2 and FIG B gives the DFA for S1.