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A language L is said to be Turing Recognizable if and only if there exist any Turing Machine (TM) which recognize it i.e., TM halt and accept strings belong to language L and will reject or not halt on the input strings that doesn鈥檛 belong to language L.

In order to prove J and $\overline{J}$both are not Turing-Recognizable,

First we will show that$A_{TM}{猢�}_{m}J$.

To prove this, we will construct TM Q

$Q=oninput\left(M,w\right)$

Write symbol 0 and $\left(M,w\right)$then in the tape and make it halt. We have:

localid="1662181827790"

So, we are able to obtain a reduction mapping of A_TM to J .

Now, we will show that$A_{TM}{猢�}_m\stackrel{炉}{J}$ . For this we will construct a Turing Machine R:
$R=oninput\left(M,w\right)$

Write symbol 1 and then $\left(M,w\right)$in the tape and make it halt. We have:

This is similar to:

So, we are able to obtain a reduction mapping of A_TM to J.

Sincelocalid="1662181817969" $A_{TM}{鈮�}_{m}J鈬�{鈮�}_m\overline{J}$,. This signifies that $\overline{J}$ is not Turing-recognizable as A_TM is not Turing-recognizable.

In same way, since. This signifies that J is not Turing recognizable.

Hence both J and $\overline{J}$are not Turing-recognizable.