Give the specification for a Turing Machine that accepts the language for $\{0$x$1$y $|13.(10$ points$)$ Yet another NP$-$Complete Problem: Set Partition

The Set$-$Partition $($SP$)$ problem takes as input a set of numbers, $\backslash ($ S $\backslash ).$ The question is whether $\backslash ($ S $\backslash )$ can be partitioned into two sets $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ where $\backslash (\backslash$mathrm$\{$B$\}=\backslash$mathrm$\{$S$\}-\backslash$mathrm$\{$A$\}\backslash )$ such that:

Show that the Set Partition problem is NP Complete.

Hint: In class we stated that the Subset Sum $($SS$)$ problem is NPComplete. In the SS problem, a set of numbers S and a number C are given, and the goal is to determine if the sum of a subset of $\backslash ($ S $\backslash )$ is exactly equal to C $.$

After showing that Set Partition is in NP$,$ try showing that Subset Sum is reducible to Set Partition. From the original set Sin the SS problem, consider adding the element:

In other words, y is the element such that when you add it to S gives the total sum of $2$C$.$