What is the minimum number of integers selected from set $S = \{1,2, 3, 4, 5, 6, 7, 8, 9\}$ such that the sum of two of the $n$ integers is even?

Question

What is the minimum number of integers selected from set $S = \{1,2, 3, 4, 5, 6, 7, 8, 9\}$ such that the sum of two of the $n$ integers is even?

my partial solution:

the sum of two of the $n$ integers is even: $\{1, 9\}, \{2, 8\}, \{3, 7\}, \{4, 6\}, \{5\}$

I don't understand if answer is $4$ or $5$ integers or other minimum number of integers.

What is the minimum number of integers selected from set $S = \{1,2, 3, 4, 5, 6, 7, 8, 9\}$ such that the difference of two of the $n$ integers is equal to $5$?

I do not know how to solve it

Answer 1

If I understand the question properly, you are meant to use the Pigeonhole Principle: if we choose any three numbers, then two of them must have the same parity (in other words, two are even, or two are odd). The sum of these two will be even.

On the other hand, if we start with only two numbers, we are not guaranteed that their sum will be even, since one could be even and the other odd.

Therefore $n=3$ is the smallest number guaranteeing that a subset of that size will have two elements whose sum is even.

Answer 2

What is the minimum number of integers selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that the sum of two of the integers is even?

The sum of two even integers is even. The sum of two odd integers is also 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 two integers such that their sum is even.

What is the minimum number of integers selected from the set $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that the difference of two of the integers is $5$?

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. Therefore, how many numbers are required to guarantee that we have two numbers whose difference is $5$?