For example, the problem *FIND-SUBSET-SUM* is in NP-equivalent. Given a set of integers, *FIND-SUBSET-SUM* is the problem of finding some subset of the integers that adds up to zero (or returning the empty set if there is no such subset). This optimization problem is similar to the decision problem *SUBSET-SUM*. Given a set of integers, *SUBSET-SUM* is the problem of finding whether there exists a subset summing to zero. *SUBSET-SUM* is NP-complete.

To show that *FIND-SUBSET-SUM* is NP-equivalent, we must show that it is both NP-hard and NP-easy.

Clearly it is NP-hard. If we had a black box that solved *FIND-SUBSET-SUM* in unit time, then it would be easy to solve *SUBSET-SUM*. Simply ask the black box to find the subset that sums to zero, then check whether it returned a nonempty set.

It is also NP-easy. If we had a black box that solved *SUBSET-SUM* in unit time, then we could use it to solve *FIND-SUBSET-SUM*. If removing one element from the set causes *SUBSET-SUM* to change its answer, then that element must be part of the solution. Just check each element this way to find the solution.