On wikipedia's article on Polynomial-time reduction it states:
Every nontrivial decision problem in P (the class of polynomial-time decision problems, where nontrivial means that not every input has the same output) may be reduced to every other nontrivial decision problem, by a polynomial-time many-one reduction. To transform an instance of problem A to B, solve A in polynomial time, and then use the solution to choose one of two instances of problem B with different answers
However I am still confused as to exactly how this works. I understand polynomial-time reductions but there are some specific questions I have in regards to this statement.
Is it required that A is non-trivial as stated on wikipedia? Some places seem to suggest that the requirement is only B has to be non-trivial, and A doesn't matter if trivial or non-trivial. Based on this question.
Can someone provide me a concrete example of this working. I don't quite understand how there are two instances of problem B with different answers and how it relies on problem A to choose