# Do problems that have unary encodings automatically become unary languages?

This problem has confused me a lot, can any of you help me out. Thank you.

• (1) Every problem has a unary encoding. (2) A unary language is a subset of $\sigma^*$ for some letter $\sigma$. No more no less. – Yuval Filmus Oct 16 at 21:29

A unary language is a subset of $$\sigma^*$$ for some letter $$\sigma$$. If your language is of this form, then it is unary. Otherwise it isn't.
A unary encoding is an encoding of some data as a subset of $$\sigma^*$$. For example, one possible unary encoding of the natural numbers is $$n \mapsto 1^n$$. Any language in which instances are unary encoded is a unary language, simply because it is a subset of $$\sigma^*$$. For example, the language of all primes encoded in unary is a unary language.