The largest number of elements possible in a subset of set  $\{1,2,\mathrm{......9}\}$ such that the sum of any two elements in that subset is unique is 

c – min. sum  $=1+2=3$ and max. sum  $=17=8+9$
Hence if 1, 2 and 8, 9 both are in subset, then maximum number of unique sums = 15
(from 3 to 17) =${ }^6{C}_2$  
But  $1+9=2+8$. Hence 6 elements is impossible. Now for 6 elements in a subset we have $\{1,2,3,5,8\}$  which satisfies given conditions $=1+2=3$