Question

Prove that A is Turing-recognizable if and only if A ≤_m A_TM.

Answer 1

Prove that A is Turing-recognizable if and only if A ≤_m A_TM.
Proof:-----------
If A is Turing-recognizable, then there exists some TM R that recognizes A. That is, R would receive an input w and accept if w is in A (otherwise R does not accept).

To show that A ≤_m ATM, we design a TM that does the following: On input w, writes (R, w) on the tape and halts. It is easy to check that (R, w) is in ATM if and only if w is in A.
Thus, we get a mapping reduction of A to ATM .