Negate the statement “If $\gcd(m,n)$ is odd, then $m$ or $n$ is odd”

Question

I need to negate this statement somehow

If $\gcd(m,n)$ is odd, then $m$ or $n$ is odd.

Does anyone know how I would do this?

Answer 1

Let $A$ be "$\gcd(m,n)$ is odd", $B$ "$m$ is odd", and $C$ "$n$ is odd". The original statement is equivalent to $\neg A \vee (B \vee C)$. To negate it, we have $$\neg[\neg A \vee (B \vee C)], \textrm{ which is equivalent to}$$ $$A \wedge \neg (B\vee C), \textrm{ which is equivalent to}$$ $$A\wedge \neg B\wedge \neg C$$ Therefore, the negation is "$\gcd(m,n)$ is odd but both $m$ and $n$ are even", which is obviously a wrong statement.

Answer 2

If $\gcd(m,n)$ is odd, then $m$ or $n$ is odd.

There is a tacit universal quantifier here:

For all values of $m$ and $n,$ if $\gcd(m,n)$ is odd, then $m$ or $n$ is odd.

The following are equivalent:

It is not true that ror all values of $m$ and $n,$ if $\gcd(m,n)$ is odd, then $m$ or $n$ is odd.

There exist values of $m$ and $n$ for which it is not true that if $\gcd(m,n)$ is odd, then $m$ or $n$ is odd.

There exist values of $m$ and $n$ for which $\gcd(m,n)$ is odd and it is not true that either $m$ or $n$ is odd.

There exist values of $m$ and $n$ for which $\gcd(m,n)$ is odd and $m$ is not odd and $n$ is not odd.