If a Language is Non-Recognizable then what about its complement?

Problem Detail: 

Is the complement of a Non-Recognizable language

1. Recognizable
2. Non-Recognizable
3. 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 "!"

1. !A(TM) is a non-recognizable language while A(TM) is a recognizable language.
2. 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.

Asked By : bongubj

Answered By : Sam Jones

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.

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Question Source : http://cs.stackexchange.com/questions/6167