help please

Question

Suppose you are to prove the statement "The difference of any two odd integers is even." Re-write this statement as a universal conditional statement. What should the first sentence of a direct proof of this statement look like?

Answer 1

Max,

 

I would restate the problem as:

 

If N1 and N2 are any two odd integers where N2 ≥ N1, then (N2 - N1) is always an even integer.

 

An aside:  Any odd integer has an even integer one away from it so any odd integer can be written as N= 2k+1 or N=2k-1 for some integer k.  For example, N=17 is odd and 17 = [2(8) + 1]  or [2(9) - 1].

 

So (N2 - N1) can be recast as [(2k + 1) - (2j +1)] for suitable k & j even or odd , and then

     (N2 - N1) = [2k - 2j] = 2(k-j)  which must be even for any integer choice of k & j.

 

That's the short proof.  You can also prove it by mathematical induction, but that would take more steps.