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Consider the following implication:

Let $x,y,z$ be integers. If exactly two of the three integers $x,y,z$ are even, then $3x + 5y + 7z$ is odd.

The contrapositive of the statement above would be:

Let $x,y,z$ be integers. If not ($3x + 5y + 7z$ is odd), then not (exactly two of the three integers $x,y,z$ are even).

I am under the impression that we can then ‘distribute’ the not like so:

Let $x,y,z$ be integers. If $3x + 5y + 7z$ is not odd, then exactly none, one, or all three of the three integers $x,y,z$ are even.

(we could replace ‘not odd‘ with ‘even’ as well.) If the aforementioned contrapositive is done incorrectly, please let me know.

One may mistake the contrapositive to instead be:

Let $x,y,z$ be integers. If $3x + 5y + 7z$ is not odd, then exactly two of the three integers $x,y,z$ are even.

In other words, we avoid the exactly two case. Now consider a different example:

Let $x,y,z$ be integers. If $xy + xz + yz$ is even, then at most one of $x,y,z$ is odd.

The contrapositive of this is:

Let $x,y,z$ be integers. If at least two of $x,y,z$ is odd, then $xy + xz + yz$ is odd.

But once again, one may mistake the contrapositive to instead be:

Let $x,y,z$ be integers. If at most one of $x,y,z$ is even, then $xy + xz + yz$ is odd.

Finally, my question is: How does one differentiate between the two examples? Intuitively, they seem counter-intuitive.

In case it may matter, I am not a native English speaker.

polite proofs
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    The simplest answer to: how do I negate a statement in words? is: don't! Convert it to symbols, and negate that instead. For example, one symbolic translation of "exactly two of $x$, $y$, and $z$ are odd" is "$(\text{$x$ odd} \land \text{$y$ odd} \land \neg(\text{$z$ odd})) \lor (\neg(\text{$x$ odd}) \land \text{$y$ odd} \land \text{$z$ odd}) \lor (\text{$x$ odd} \land \neg(\text{$y$ odd}) \land \text{$z$ odd})$; now negate that. The result can then be 'streamlined' into words, but the surest way to check your understanding is symbolically. – LSpice Jan 10 '21 at 23:59
  • @LSpice I see your point, but my book declared these results and I was supposed to understand why they are the way that they are intuitively. – polite proofs Jan 11 '21 at 00:03
  • The point is that the negation of "exactly two are blah" is not "exactly two are not blah", as you propose; nor is it "exactly zero, one, or three are not blah", which you correctly say it is mistaken; it is rather "exactly zero, one, or three are blah". I would prefer to check this symbolically, but, from the intuitive point of view, some number are blah; and the only way that number is not two is (in your example) if it is zero, one, or three. (Consider, for example, the numbers 0, 2, 4, of which neither "exactly two are odd" nor "exactly two are even" is true.) – LSpice Jan 11 '21 at 00:04
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    @LSpice Just to be clear, you are claiming that the contrapositive of: "Let $x,y,z$ be integers. If exactly two of the three integers $x,y,z$ are even, then $3x + 5y + 7z$ is odd." is "Let $x,y,z$ be integers. If $3x + 5y + 7z$ is even, then exactly none, exactly one, or exactly three of $x,y,z$ are odd."? – polite proofs Jan 11 '21 at 00:11
  • No, that is not the correct contrapositive. – LSpice Jan 11 '21 at 00:22
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    Sorry, then it would be "Let $x,y,z$ be integers. If $3x + 5y + 7z$ is even, then exactly none, exactly one, or exactly three of $x,y,z$ are even." – polite proofs Jan 11 '21 at 00:23
  • Yes, that is the correct contrapositive (although I can't guarantee it's how your instructor wants it written; for example, they may prefer "$3x + 5y + 7z$ is not odd"). – LSpice Jan 11 '21 at 00:25
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    @LSpice No instructor here; just self-studying to learn math. Thank you for patience. – polite proofs Jan 11 '21 at 00:26
  • No problem. This is a difficult topic, and it's important to get it right. Your current draft of the post proposes that the negation of "exactly two of the three integers $x$, $y$, $z$ are even" is "exactly two of the three integers $x$, $y$, and $z$ are even", which I think is not what you mean. – LSpice Jan 11 '21 at 00:32
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    @LSpice Sorry, not really used to editing on here. I edited it now. – polite proofs Jan 11 '21 at 00:43

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