How-to determine whether a given set is recursively enumerable?

Question

I'm stuck with this problem I have to solve.
Set $A = \{ x | \Phi_{x}(x): defined \}$
Set $B$ is produced from set $A$ by taking out all even numbers.
Is set $B$ r.e.? How does one prove that?

Answer 1

So $B$ is all odd numbers in $A$, just start enumerating $A$, that is preforming the succesive calculations untill they converge and then jut take the odd numbers.

Answer 2

There are two useful ways to think about how a set $S$ is recursively enumerable.

1. $S$ is recursively enumerable if there is an algorithm that lists its elements.
2. $S$ is recursively enumerable if there is an algorithm which, given an element of $S$, halts and says “yes”, and which given some other value either halts and says “no” or runs forever. (In this case we say that the algorithm “semidecides” $S$.

Let's use just #1. We are given that $A$ is RE. Then there is an algorithm $M_A$ that lists the elements of $M$. We would like to use this to build an algorithm $M_B$ that lists only the elements of $B$. $M_B$ should contain a copy of $M_A$, examine the outputs of $M_A$, and for each one… ?

Now let's use #2. We are given that $A$ is RE. We are given an algorithm $M_A$ that semidecides $A$: given any element $a$ of $A$, it will halt and say “yes”, and given any other element it will either halt and say “no” or else run forever. We would like to use this to build an algorithm $M_B$ that semidecides $B$: given any element of $b$ of $B$ it must halt and say “yes”, and given anything else it must either halt and say “no” or else run forever. $M_B$ may contain a copy of $M_A$, and when $M_B$ gets some input $x$, it has the option to ask $M_A$ what it thinks of $x$; if $M_A$ halts, then $M_B$ can use $M_A$'s answer in choosing what to do. What should $M_B$ do in order to correctly semidecide whether $x$ is in $B$?