Karp's reduction strategy One way to prove that a problem $X$ is NP-Complete is to pick two NP-Complete problems $Z$ and $W$ and show that ($\leq_p$ is polynomial reduction):
$$\begin{array}{rr}X \leq_p Z &amp; (1)\\W \leq_p X&amp;(2)\end{array}$$
The usual way to prove NP-Completeness however, is to follow Karp’s strategy which consists in proving that
$$\begin{array}{rr}X\in \textsf{NP} &amp; (3)\\W \leq_p X&amp;(4)\end{array}$$
I am trying to show that $(1)$ and $(3)$ are equivalents, but but I'm not sure where I can start. Can someone help me with how to do this?
$(3)\Rightarrow (1)$ should just be a consequence of the definition of $\textsf{NP}$-completeness.
For $(1)\Rightarrow (3)$, you can use the fact that there exists a function $f$, computable in polynomial time, such that $u\in X \Leftrightarrow f(u) \in Z$. Now to decide if $u\in X$, you can compute $f(u)$, and then decide nondeterministically if $f(u)\in Z$.