According to the statement of Pumping Lemma:-
For any regular language $L$, there exists an integer $n$, such that for all $x ∈ L$ with $|x| ≥ n$, there exists $u, v, w ∈ Σ^{*}$, such that $x = uvw$, and
$(1) |uv| ≤ n$
$(2) |v| ≥ 1$
$(3) for \ all \ i ≥ 0: uv^{i}w ∈ L$
In Simple terms , it doesn't matter how many times sub string $v$ is pumped , the string would still belong to the language ,and if we can find some string $uvw$ such that on pumping $v$ , the string is not in the language then we can surely say that the language is not regular. Pumping Lemma is a negative test.
Let's take the 1st Language:-
$A1=\{0^{n}1^{n}2^{n}|n≥0\}$.
Let's take a string $w="001122" \in A1$.
In this string , let $u=00 , v=11,w=22$.
So according to pumping lemma , it doesn't matter how many times the string $v$ is pumped , the resulting string would still be in the language .
Let's pump $v$ once.
Resulting string= $"00111122"$.
But this string doesn't belong to $A1$ . Thus pumping lemma fails here , thus we can say that the language is not regular.
Taking the next Language .
$A2=\{www|w∈{a,b}^{*}\}$.
Let us take a string $w=ababab$.
In this string , let $u=ab , v=ab,w=ab$.
So according to pumping lemma , it doesn't matter how many times the string $v$ is pumped , the resulting string would still be in the language .
Let's pump $v$ once.
Resulting string= $"abababab"$.
The resulting string doesn't belong to the Language , thus pumping lemma fails here , thus the language is not regular.
Taking the third one.
$A3=\{a^{2{n}}|n≥0\}$.
Let us take a string $w=aaaa$ which is $a^{2^{2}}$.
In this string , let $u=a , v=aa,w=a$.
So according to pumping lemma , it doesn't matter how many times the string $v$ is pumped , the resulting string would still be in the language .
Let's pump $v$ once.
Resulting string= $"aaaaaa"$.
But this string doesn't belong to $A3$ . Thus pumping lemma fails here , thus we can say that the language is not regular.
