NP-Complete and Poly Time Reduction Problems

Question

I Took Some Priminlairity Learning Method on Complexity Theory. I get trouble with some definition. anyone could help me, Why the mentioned statement is True?

if a Problem A can be reducible to Problem B in Poly time ( A $<_p$ B).

a. if B can be solve in O(nlogn), so then A can be solved in O(nlogn).

b. if B can be solved in poly time and A be NP-Complete, then All of NP class problems can be solved in poly time.

Answer 1

(b) is true because:

1. $A$ is NP-complete, so (by definition) any problem $X$ in NP can be polynomial-time reduced to $A$.
2. $A$ can be polynomial-time reduced to $B$, by hypothesis
3. $B$ can be solved in polynomial time, by hypothesis

so an algorithm for solving $X$ in polynomial time is: reduce $X$ to $A$, then to $B$; the composition of these two polynomial-time reductions requires polynomial time. Then use the algorithm for $B$.

As I said in the comment, I think (a) is false.

Answer 2

a. is not true, see @MJD's comment.

b. is true because if A is some NP-Complete problem that can be reduced in polynomial time to B, and B can be solved in polynomial time, then A can also be solved in polynomial time: take polynomial time to reduce A to B, and take polynomial time to solve B. Since poly time + poly time = poly time, A is solved in poly time. Since A is NP-Complete, every problem in NP can be reduced to A in polynomial time, by definition. So every problem in NP can be solved in polynomial time be reducing to to A, then reducing to B.