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.

(a) We know that , sum of two integers is even iff both are odd or both are even.

On the other hand, if we select an even integer and an odd integer, their sum is odd, so two integers is not sufficient.

If we take three integers from set $S,$ then at least two of them must have the same parity, which guarantees that we will have even sum.

Answer: three integers.

(b) Consider the subsets $\{1,6\}, \{2,7\}, \{3,8\}, \{4,9\}, \{5\}.$ The worst case scenario is that we select one number from each of these subsets.

Thus, it is possible to select five integers and avoid a solution to $x-y=5,$

 from this selection, but not six.

Answer: six integers.