If a Language is Non-Recognizable then what about its complement? Is the complement of a Non-Recognizable language


Recognizable
Non-Recognizable
May be Recognizable, May be Non-recognizable. I.e cant comment.


A mathematical proof would be of great help since im unable to think of any way to prove this.

I did some research on this and found below examples. Im specifying complement using a "!"


!A(TM) is a non-recognizable language while A(TM) is a recognizable language.
EQ(TM) is a non-recognizable language and !EQ(TM) is also non-recognizable language


The above two would mean that we simply cant comment on the Recognizability of the complement of non-recognizable language. But I feel that there should be some way to prove (or disprove) this.
Assuming that you can prove statement 1 and 2 above, you have just presented a proof. You're trying to show that there exists a language which is non-recognizable such that its complement is recognizable. You then prove that $!A(TM)$ has this property. Then you want to show that there exists a non-recognizable language whose complement is also non-recognizable and then you show that $EQ(TM)$ has this property.

The point is that to prove that you can't comment, you need only to show that both possibilities are possible ie. that examples of each exist.