# Explain the difference between Turing Complete and Turing Equivalence

I'm not sure if I understand the difference between Turing Complete and Turing Equivalent programming languages.

A computational system that can compute every Turing-computable function is called Turing-complete (or Turing-powerful).

A Turing-complete system is called Turing-equivalent if every function it can compute is also Turing-computable; i.e., it computes precisely the same class of functions as do Turing machines.

I'm not sure if I understand the difference between these two terminologies properly and what "Turing-computable" means. So for example if I have a programming language $$X$$, it is Turing complete if it can do any and everything that a Turing Machine can do? And it $$X$$ ends up being Turing complete and it can be shown that all of the functions that it computes are Turing computable, then $$X$$ is also Turing equivalent?

• There exist functions that can't be computed by TM. You can imagine some computational system that can magically compute more functions (e.g. Oracle machine). This system won't be Turing-equivalent. – user114966 Jan 12 at 15:46