THREE$2$ONE$-$PARTITION Problem :

Instance: A finite set of positive integers Z$=\{$z$1,$z$2,...,$zn$\}.$ Question: Is there a subset Z$\text{'}$ of Z such that

Sum of all numbers in Z$\text{'}=3$ Sum of all numbers in Z$-$Z$\text{'}$

SUM$-$OF$-$SUBSETS Problem

Instance: A finite set A $=\{$a$1,$a$2,...,$am$\}$ and M$.$ Question: Is there a subset A$\text{'}$ in A s$.$t$.$ ai $=$ M$?$

ai in A$\text{'}$

Given that THREE$2$ONE$-$PARTITION Problem is NP$-$Complete, prove that the SUM$-$OF$-$ SUBSETS Problem is NP$-$Complete by reducing THREE$2$ONE$-$PARTITION Problem to it$.$

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$($a$)$ Give a nondeterministic polynomial time algorithm for the SUM$-$OF$-$SUBSETS Problem. $($Use Guess statements in your solution, e$.$g$.$ Guess$(\{0,1\})$ returns $0$ or $1)$

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$($b$)$ Define the transformation from the THREE$2$ONE$-$PARTITION Problem to the SUM$-$OF$-$ SUBSETS Problem and give the if$-$and$-$only$-$if proof.