<Closure Property>
Regular Language is closed under union, intersection, concatenation, star, and complement
Context Free Language is closed under union, concatenation, and star. But, it is NOT closed under intersection and complement.
Decidable Language is closed under union, intersection, concatenation, star, and complement.
Recognizable Language is closed under union, intersection, concatenation, and star. But, it is NOT closed under complement.
<Decidability>
All finite languages are decidable
Only infinite language can be undecidable.
There are many infinite languages that are decidable (Σ* is infinite but decidable)
<Some Facts>
Every finite language is always regular language.
Every subset of a countable set is either countable or finite.
<Decidable language vs Recognizable language>
Decidable language is also known as Recursive language. Turing machine will always halt for decidable languages. It will accept and halt if it is in the language and reject (= halt) strings if it is not in the language.
Recognizable language is also known as Recursively enumerable language. Turing machine will halt sometimes & may not halt sometimes. It will accept and halt if it is in the language. It will either reject (= halt) or not halt if it is not in the language.