I'm just a little confused about the wordings for reductions and would appreciate some help as I am very new to reductions.
I have a question where I have two sets, A and B, and I have to provide a reduction A $\leq_p$ B and then I have to prove that the reduction is correct.
What does this mean exactly?
For the reduction portion do I have to:
Provide some sort of explanation of B in terms of A?
Provide some sort of explanation of A in terms of B?
Both 1 & 2?
And then I'm assuming to prove the reduction is correct, I need to show A $\implies$ B as well as B $\implies$ A?
(The exact question has A as the set of un-directed graphs with a Hamiltonian path and B is the set of un-directed graphs with a Hamiltonian cycle)
Any clarification would be greatly appreciated!!