Faid L.

asked • 02/12/18

Contradiction prove question, "Prove by contradiction that the difference between any odd integer and any even integer is odd"

How to do this, please help me. Thank you very much!

1 Expert Answer

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Andrew M. answered • 02/12/18

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Even integers will be of the form 2x
odd integers will be of the form 2y+1
 
Difference:  2y + 1 - 2x
                = 2y - 2x + 1
                = 2(y-x) + 1
 
By definition 2(y-x) will be even and thus 2(y-x) + 1 will be odd
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Andrew M.

Actually, for the proof by contradiction, we first assume the opposite is true and
then show that the opposite creates a contradiction.  Thus we must try to show
that the difference between an odd integer and an even integer will be even.
 
Odd integer:  2x + 1 
Even integer:  2y
 
2x + 1 - 2y = 2x - 2y + 1 = 2(x-y) + 1
Since 2(x-y)+1 will be odd, this contradicts the assumption, and thus
shows that it is untrue.  Therefore, the difference between and odd
integer and an even integer will be odd.
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02/12/18

Faid L.

Ah i understand clearly, thank you Andrew M.
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02/12/18

Andrew M.

You're welcome Faid.  Best of luck in your endeavors.
:-)
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02/12/18

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