Prove disprove an existence of mapping reduction between 2 sets I am currently studying mapping reduction in computational theory and finding it hard to grasp the concept fully. 

For reference, consider the following given WHILE-Prog sets:

A = { (p.d) | p doesn't halt on input d } = Complementary HALT set.
B = { p | p halts on exactly one input (the input is unknown) }

Is A &lt; B. meaning, is there a mapping reduction from A to B?



Can someone suggest a hint? 

knowing that A belongs to coRE did not help much. 
B doesn't seem to belong to either RE nor coRE.
One basic idea is to observe that $B$ looks harder than $A$ since


we need to run a given Turing machine on one given input in order to find whether the pair is in $A$ but
we may need to run a given Turing machine on infinity many inputs in order to find whether it is in $B$.


That observation should motivate us to find a reduction from $A$ to $B$.



Suppose $(p,d)\in A$. Construct a machine that behaves as the following.

Upon an input string $s$, it checks whether $s$ is $d\sigma$, where $\sigma$ stand for a fixed letter in the alphabet.


If yes, it halts. 
If no, it checks whether $s$ is $d$. 


If no, it loops forever. 
If yes, it simulate $p$ on $d$.



It should be routine to verify the construction above is indeed a reduction from $A$ to $B$.