Answer: (iv) All of these

Explanation:

Let $A \le_p B$ denote polynomial-time many-one (Karp) reduction.

• (i) If $A \le_p B$ and $B \in P$ then $A \in P$: True. Compose reduction $f$ (poly-time) with poly-time decider for $B$ → poly-time decider for $A$. Hence $P$ is closed under $\le_p$.

• **(ii) If $A \le_p B$ and $A
  otin P$then$B
  otin P$**: True — contrapositive of (i). If$B$were in$P$, (i) would put$A$in$P$, contradiction.

• (iii) If $A \le_p B$ and $B \le_p C$ then $A \le_p C$: True — $\le_p$ is transitive; compose the two poly-time reduction functions (polynomial of polynomial is polynomial).

All three are fundamental lemmas used in NP-completeness proofs.

Correct choice: (iv) All of these