A_{TM} = {⟨M,w⟩| M is a TM and M accepts w} is decidable

The complement of A_{TM} is Turing-recognizable 

A partially decidable language is also known as 

A_{TM} = {⟨M,w⟩| M is a TM and M accepts w} is Turing recognizable

A_{DFA} = {⟨B, w⟩| B is a DFA that accepts input string w} is an undecidable language. 

A set A is countable only if A is finite 

which of the following problems is undecidable 

A language is decidable if and only if it is Turing-recognizable

knowing when a problem is algorithmically unsolvable is useful because

Every Turing recognizable language is also decidable