Classifying NP Completeness and Hardness

Choose the correct statement(s):

• (A) If X is an NP-complete problem, then X is an NP problem
• (B) If X is an NP-complete problem, then X is an NP-hard
• (C) Let X be an NP-complete problem. If X can polynomial reduce to a problem Y, then Y is an NP-complete.
• (D) Let X be an NP-complete problem. If Y can polynomial reduce to a problem X, then Y is an NP-complete.
• (E) Let X be an NP-complete problem. If X can polynomial reduce to a problem Y, then Y is an NP-hard.

My answer is (A)(B)(C)(E):

• (A)(B) : X belongs to NP-complete, means X belongs to NP and NP-hard
• (C) true
• (D) Y may be P, NP-hard or NP-complete
• (E) Y is an NP-complete, and it also is an NP-hard

Is answer true?

Here are some corrections:

(C) False. Y is NP-hard, but not necessarily in NP.

(D) False. Y is in NP, not necessarily NP-complete.

(E) True, but Y is no necessarily NP-complete.