Language, L = {aⁿbⁿ | n ≤ 10¹⁰}, check whether this language is regular or not

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From the definition is pretty clear that language is finite language as the bound is 10¹⁰, which is a finite number.

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Language, L = {aⁿbⁿ | n ≥ 1}, check whether this language is regular or not

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We can see that for this language we cannot create FA because it needs infinite memory to remember each occurence of 'a' and 'b'.
So we say that language is not regular.

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Language, L = {ww^R | |w| = 2} and Σ = {a,b} check whether this language is regular or not

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Note:Here ww^R means a string w and then its reverse..
From the definition is pretty clear that language is finite language as the bound is |w| = 2, which is a finite.

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Language, L = {ww^R | w ε (a,b)^*} and Σ = {a,b} check whether this language is regular or not

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Language is inifinite but we cannot create FA as bba|abb, we have to remember bba so same thing for infinite string which tells we should have infinite memory.
So not regular.

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Language, L = {ww | w ε (a,b)^*} and Σ = {a,b} check whether this language is regular or not

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Note:Here ww means a string w and again the same string.
Language is inifinite but we cannot create FA as bba|bba, we have to remember bba so same thing for infinite string which tells we should have infinite memory.
So not regular.

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