Recursive Language
For a given language if a turing machine can be designed which will halt on below 2 decisions:
- Whether String belongs to language
- Whether String does not belong to language
Below picture will put more light on the explanation:

In above pic the outer rectangle is Σ*(all possible strings over given input) and langauge is a subset of it.
Recursive Enumerable and Recursive Langauges
Recursive languages are subset of Recursive Enumerable Langauges.Means: Every recusrive language is recursive enumerable language but vice-versa is not true.

Recursive Langauge Closure Properties
Closure Property | Status |
---|---|
Union | Yes |
Intersection | Yes |
Set Difference | Yes |
Compliment | Yes |
Intersection with Regular Language | Yes |
Concatination | Yes |
Kleen Closure | Yes |
Kleen Plus | Yes |
Reversal | Yes |
Homomorphism | No |
ε-free Homomorphismx | Yes |
Inverse Homomorphism | Yes |
Substitution | No |
ε-free Substitution | Yes |