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**1.From the problem statement**

Let Z

Z

Define Z

Let Z

^{+}be the set of all positive integers; that is,Z

^{+}= {1,2,3,...}Define Z

^{+}x Z^{+}= {(a_{1}, a_{2}) : a_{1}is an element of Z^{+}and a_{2}is an element of Z^{+}}**2. If S is contained in Z**

^{+}and |S| >=3, prove that there exist distinct x,y that are elements of S such that x+y is even.**3. I need some help on where to start as I'm quite lost.**

I can see how this would be true. for example the set S = {1,2,3,4}

The pairs (1,3) and (2,4) both satisfy the condition of x+y being even.

I can see how this would be true. for example the set S = {1,2,3,4}

The pairs (1,3) and (2,4) both satisfy the condition of x+y being even.

the are extensions of this homework problem as well, but I'm having trouble with the first bit...