Find the minimum number n of integers to be selected from S = {1, 2,..., 9} so that: (a) The sum of two of the n integers is even. (b) The difference of two of the n integers is 5

The give set is

$S=\{1,2,3,4,5,6,7,8,9\}$

$S$ contain 5 odd number and 4 even number.

We know that , sum of two integer is even iff both are odd or both are even .

The minimum number n of integer to be selected from $S$ ,so that  The sum of two of the n integers is even is 2 .

( since , take a set of 2 odd or even integer ).

If the difference of two integer (say )$n_1,n_2$ is 5 then $n_1-n_2=5$

$\implies n_1=5+n_2$ .

Therefore the minimum number n of integer selected from S will be 2,so that the difference of two integer is 5.

For example : Take the set $\{ 1,6\}$ .