The subset$-$sum problem is defined as follows: Given a set of positive integers, S $=($a$1,$ ag$,$ as$,...,$ a$)$ and positive integer to$,$ is there a subset of s whose elements sum to w$?$ A dynamic program for solving this problem uses a $2-$dimensional boolean array X $[$i$,$ j$],$ with n rows $1<=$ i $<=$ n and w $+1$ columns $0<=$ j $<=$ w$.$ X $[$i$,$ j$]$ is TRUE if and only if there is a subset of $($a$1,$ a$2,...,$ a$,$ whose elements sum to j$.$ Which of the following is valid for $2<=$ i $<=$ n and $<=$ w$?$