Are the following sets solvable (respectively,enumerable)?
a) The set {$xy$ | $x ∈ A$, $y ∈ B$}, where $A$ and $B$ are solvable (enumerable).
b) The set $A ⊆ B$, where $B$ is solvable (enumerable).
As for $a$, I want to say that if $A$ and $B$ solvable we can built algorinm
An algorithm like if an element belongs to both $A$ and $B$, then it belongs to our set, otherwise it does not belong, please correct how it looks in the correct form and how to take on the rest of the items
a)I assume the operation in a) is product in which case the set is solvable/recursive and recursively enumerable respectively. The reason is that given that we have at hand the two members of the sets it is an easy task to take their product(in fact we can do this in polynomial time as a function of the size of the inputs), just think about the fact that your hand held computer can do that in no time given enough room on it's screen.
b)Regarding b however things are not the same. You must be familiar with the problem which asks if the set of halting turing machines is solvable provided we know how that description should look like. Well let's take a set $B = \{e|\text{e is the description of a Turing machine}\}$. This set is solvable. Now the halting set is a subset of B which is usually written as K so we have that $K\subset B = \{e\ |\ \text{e is a description of a Turing machine that halts on at least one input}\}$ which we know is not solvable since the days of Turing. A similar situation occurs If the original set is enumerable, we just take a particular subset of K for exemple which is enumerable but it's subset: $K^* = \{e\ | \ \text{e is the description of a Turing machine that halts on all inputs}\}$ is not enumerable.
