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I need to write the following sentence in the form "if, then": "The only possibility for the integer n - 3 to be even is for n to be odd"

The textbook in which I found this question gives the solution as: "If n is an odd integer, then n − 3 is even"

However, I struggle to understand how these two statements are equivalent since the first statement is false when n - 3 is even and n is even whereas the second one is true in this case. My answer to this question was the converse of the manual solution: "If n - 3 is even, then n is an odd integer".

Am I making a mistake in my reasoning or is there an error in the textbook ?

raphiki
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    You are right. – Anne Bauval Oct 12 '22 at 21:43
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    Thanks! I saw the same solution on an independent website so I was not too confident... – raphiki Oct 12 '22 at 21:48
  • The textbook is wrong – Dark Rebellion Oct 12 '22 at 22:03
  • I also agree with your answer. In fact, I hate when technical writing is obfuscated like this. It slows down comprehension when reading and it increases the possibility of being misinterpreted. In my own writing I even try to avoid what I call a backwards "if ... then" construction (see end of this answer), although I'm sure I've slipped up from time to time. On the other hand, it is important to be able to understand natural language usage (where mixing things up decreases exposition dullness), so such exercises do have a purpose I suppose. – Dave L. Renfro Oct 12 '22 at 22:08
  • "the first statement is false when $n-3$ is even and $n$ is even": this situation never arises. Indeed, part of why this is a confusing question is that both your solution and the manual's solution are true implications: $n-3$ is even if and only if $n$ is odd. – Greg Martin Oct 12 '22 at 23:13

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