# Proving the singleton language {x} is regular for all x ∈ Σ*

So I'm aware that the singleton language is in fact regular for all x ∈ Σ*, but I do not understand why it is. A formal proof would help, but also getting some intuition as to why it is regular would also be appreciated! As of now I'm just aware of it as a property, but I don't have a good grasp on why it is regular.

• Any finite language is regular. – Derek Elkins Feb 12 at 2:57
• What is your definition of a regular language? If it is a language accepted by a DFA, then you can easily construct a DFA accepting $\{x\}$, which has $|x|+2$ states. It's a good exercise for you. – Yuval Filmus Feb 12 at 5:02
• If your definition of regularity is based on regular expressions, then this is a very basic fact. Otherwise, this could be helpful. – dkaeae Feb 12 at 8:49
• I thought x ∈ Σ* isn't assuming that x is finite? Like, x could be an infinitely large string. So {x} would be a single string with infinite length. – James Swanson Feb 12 at 14:21

$$\Sigma^*$$ is a set of finite strings. There are infinitely many of them, but each one is finite – just like there are infinitely many natural numbers but each one of them is finite.

So, for any $$x\in\Sigma^*$$, $$\{x\}$$ is a finite language (it contains one string) of finite strings (because everything in $$\Sigma^*$$ is finite), so it's regular.