Let be a set of four distinct integers. Prove that if either (1) for each , the integer and the sum of any 2 of the remaining 3 integers of S are of the same parity, or (2) for each , the integer and the sum of any 2 of the remaining 3 integers of S are of different parity, then every pair of integers S are of the same parity.

Here is my attempt at proof by Contrapositive:

Let P: for each , the integer and the sum of any 2 of the remaining 3 integers of S are of the same parity.

Q: for each , the integer and the sum of any 2 of the remaining 3 integers of S are of different parity.

R: every pair of integers S are of the same parity.

In logic symbols:

Contrapositive:

So assume : there exists a pair of integers of S of different parity. And show that follows.

: there exists , such that the integer and the sum of any 2 of the remaining 3 integers of S are of the different parity.

: there exists , such that the integer and the sum of any 2 of the remaining 3 integers of S are of the same parity.

Without loss of generality, let a and b be the pair of integers with different parity and without loss of generality let a be even and b be odd. Then for some and for some .

Case 1: Both remaining c and d are even. Then for some and for some .

For , the sum of any 2 other integers will be of opposite parity because they are all even. So is satisfied.

Here is where I'm stuck, to show , I need an integer such that, the sum of any 2 of the remaining 3 integers have the same parity. But I can't have that because if I choose one of the even numbers, then I eventually have to select an odd number to be in the sum of the 2 of the remaining 3.

Case 2: Both remaining c and d are odd.Then for some and for some .

Same problem as case 1 except reversed. I can show because for , the 3 remaining are all odd numbers, so adding any 2 of those 3 will give an even number. So they have the same parity.

But how do I show because if I choose an odd number, adding an odd number with an even number will still give an odd number, so the sum of those 2 will end up having the same parity again.

Case 3: c and d are of different parity. Without loss of generality let c be even and d be odd. Then for some and for some .

Case 3 I can't show either.

I don't understand what I'm doing wrong. Any suggestions would be appreciated.