# $a^{2n}$ where $n\geq1$ is not Regular using Pumping Lemma [duplicate]

I'm having trouble understanding Pumping Lemma for $RL$ , i know that $0^{2n}$ is regular and we can design a FSM for it.
Here's my pumping Lemma statement

Let $p$ be the pumping length and my string is $0^{2p}$
1. $z=|0^{2p}|=> 2p > p$ (Satisfied)
2. $z=uvw$ such that $|uv|\geq p$, I'm choosing $|u|=n-p$ and $|v|=p$ so that $|uv|\geq p$ (Satisfied)
3.Now $u(v^i)w$, I choose $i=3$ then $|uvw|=2(n+p)$ which is even
But if $n=2$ then $|uvw|=2n+p$, how do i know if this could be even/odd?

• You don't get to choose the partition $z = uvw$. The pumping lemma only guarantees that such a partition exists. – Yuval Filmus Aug 23 '17 at 8:25
• Haven't we had this or very similar languages a lot recently? – Raphael Aug 23 '17 at 11:32
• Title says "...not Regular using Pumping Lemma", the body text says "...i know that $0^{2n}$ is regular...", the question is not clear. – fade2black Aug 23 '17 at 14:09
• @fade what I meant was we could prove it is not regular but actually it is regular – user2643191 Aug 23 '17 at 14:17
• Okay, I see your point. Please have a look at this almost identical question if it helps. – fade2black Aug 23 '17 at 14:28

You can't prove that a language is regular by applying the pumping lemma.

Regular Languages are a subset of Languages that satisfy the pumping lemma. So you can only proof irregularity with the lemma. (By assuming that the language is regular and coming to a contradiction)

One way to show that a language is regular, is e.g. constructing a Regular Expression, NFA or DFA (all of these being equivalent).

• There are other ways of proving that a language is regular, for example using closure properties. – Yuval Filmus Aug 24 '17 at 8:41
• @YuvalFilmus thank you, I edited my answer. – KillPinguin Aug 24 '17 at 8:46