I am completely lost on how to do this.

Prove that an integernis evenif and only ifn+1is odd

thanks in advance for any help

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- Oct 18th 2009, 04:40 PMsprint10sProofs can't figure it out
I am completely lost on how to do this.

Prove that an integer*n*is even**if and only if***n+1*is odd

thanks in advance for any help - Oct 18th 2009, 05:26 PMpickslides
For starters make n+1 odd as follows

2(n+1)+1 now show the previous number is even. - Oct 18th 2009, 05:30 PMoldguynewstudenteven odd proof
T.P. n is even iff n+1 is odd

Assume n is even then let n = 2k for some k element of Z (definition of even)

Then n+1 = 2k + 1, but 2k + 1 is the definition of an odd number. This proves the forward direction now reverse it.

Let n + 1 be odd then by definition of an odd number n + 1 = 2k + 1 for an arbitrary k element of Z

Now subtract 1 from both sides and you have n = 2k which by definition is even.

This completes the proof in both directions.