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I am wondering what the negation of the statement: "$y$ is a composite number smaller than two" would be. Would it be:

1) $y$ is a composite number greater than or equal to two

or is it,

2) $y$ is a non-composite (prime) number greater than or equal to two?

Thanks.

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  • $\begingroup$ $\neg( A \land B) = \neg A \lor \neg B$. $\endgroup$
    – basket
    Sep 9, 2016 at 2:05
  • $\begingroup$ There is a subtle difference between a "syntactical" and a "semantical" negation in this case. By De Morgan's law stated above the synctatical negation is "$y$ is $\geq 2$ or it is a prime number". But if it is clear that $y$ ranges among the positive integers, since there is no composite number smaller than $2$, a semantical negation may be a rose is a rose or any tautology. $\endgroup$ Sep 9, 2016 at 3:57

3 Answers 3

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The negation is formed by simply inserting not:

y is not a composite number smaller than two

You're probably asking how to apply De Morgan's Law to this new statement, in which case you're looking for

y is not a composite number OR y is not smaller than two

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    $\begingroup$ Even safer is prepending “It is not the case that,” so the negation (same in this case) would be “It is not the case that $y$ is a composite number smaller than $2$.” Observing that the original statement, “$y$ is a composite number smaller than two” means “$y$ is a composite number and $y$ is smaller than two,” the negation can be worked out with rules of propositional logic and becomes “Either $y$ is not a composite number or $y$ is not smaller than two.” $\endgroup$
    – Steve Kass
    Sep 9, 2016 at 2:29
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Either y is not a composite number or y is not smaller than 2

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You should think in sets. Let $A$ be the set such that y is composite and smaller than two. $A^c$, which is the negation of $A$, is the set such that $y$ isn't composite or $y\ge 2$.

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