The last problem in this section!

**The problem: **Integer Expression Membership. This is problem AN18 in the appendix.

**The description: **Given an integer expression e over the expressions ∪ and +, which work on sets of integers. The ∪ operation unions two sets of integers together, and the + operation takes two sets of integers and creates the set of sums of elements from each of the sets. Given an integer K, is K in the set of integers represented by e?

**Example: **Suppose A = {1,2,3}, B = {4,5,6}, C = {1,3,5}.

Then the expression (A∪C) would be {1,2,3,5} and the set (A∪C) + B would be {2,4,6,3,5,7,6,8,10}. So if we were given that expression and a K of 8, a solver for the problem would say “yes”, but if given a K of 9, the solver for the problem would say “no”.

**Reduction:** Stockmeyer and Meyer show this in their Theorem 5.1. They reduce from Subset Sum (they call it Knapsack, but their version of Knapsack doesn’t have weights, so it’s out Subset Sum problem). So from a Subset sum instance (a set A = {a_{1}, …, a_{n}} and a target B), we build an expression. The expression is (a_{1} ∪ 0) + (a_{2} ∪ 0) + … + (a_{n} ∪ 0). Then we set K=B.

Notice that the set of integers generated by this expresion is the set of all sums of all subsets of A. Asking is B is a member is just asking if there is a way to make the above expression (for each element a_{i}, either taking it or taking 0) that adds up to exactly B. This is exactly the Subset Sum problem.

**Difficulty:** 4. The reduction itself is easy. It might be hard to explain the problem itself though.