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Let's say I want to express this statement using quantifiers : "For every two odd numbers, the sum of them is even."

$\forall x,y \in \mathbb{Z}$ ( $(O(x)$ AND $O(y)$) $\rightarrow E(x+y) )$

where $O(x)$ means $x$ is odd, and $E(x)$ means $x$ is even.

My question is, is the expression correct ? I think there may be an error whenever either $O(x)$ or $O(y)$ is false, since it will make the whole implication correct. But I just don't know how to express it in another way.

Thank you.

BLAZE
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user279396
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1 Answers1

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It is correct. You're right that, if (say) we take $x=2$ and $y=3$, the inner implication winds up being true - but that's a good thing: it means "what if $x=2$ and $y=3$?" isn't a counterexample to the statement (it's not a counterexample because $x$ isn't odd - it's like if you said, "There are no consecutive primes bigger than 3" and I said, "What about 2 and 3?").

Noah Schweber
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