Proof that $L=\{a^ncb^n| n \in \mathbb{N}\}$ is not regular
Proof:
Lets assume L is regular. Then we know that L must meet the requirements of the pumping lemma.
So let p the pumping number.
Let $w=a^pcb^p$. $w$ is obviously of the length p and is in L. Therefore it should be possible to split w into three pieces xyz such that $|y|&gt;0,|xy|\leq p,xy^iz$ is in L $\forall i \in N$.
Because $|xy|\leq p$ $y $ can only contain the symbol $a$(If $y$ would contain a symbol different from a it would imply that $|xy|&gt;p$, which is not possible). Therefore $y$ must be in the form $y=a^{p-k},0\leq k&lt;p$. So the word w equals $w=a^ka^{p-k}cb^p$, if we set i=2 we get $a^ka^{2p-2k}=a^{2p-k},0&lt;=k&lt;p$ and because $ a^{2p-k}\neq b^p$ it follows that the pumped $w$ is not  in $L$. Which is a contradiction. Therefore $L$ is not regular.
$\tag*{$\blacksquare$}$