Question

Show that A is Turing-recognizable $A_m\le A_{TM}$

Short answer

It is proved that A is Turing Recognizable.

Step-by-step solution

Introduction of Turing Recognizable

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’t belong to language L .

To show A is Turing recognizable

Before we proceed, we know that A_TM is Turing Recognizable.

Let us assume that A is Turing Recognizable, this means we can construct a Turing Machine M for A , such that $L\left(M\right)=A$.

Assume f is Mapping Reduction Function which is defined as:
$f\left(w\right)=\left(M,w\right)$

So,$w\in A$if and only if$\left(M,w\right)\in A_{TM}$.

So, if, A_m$\le$A_TMthen A must be Turing Recognizable as A_TM is Turing Recognizable