Writing converse of a statement

Question

I read the following statement

For any nonzero integers $a$ and $b$ there exist integers $s$ and $t$ such that $\gcd(a,b)=as+bt$.

and then tried to think about its converse, which I thought is

For any two nonzero integers $a$ and $b$ if there exists integers $s$ and $t$ such that $as+bt=d$ then $d=\gcd(a,b)$.

Is the converse right?

Answer 1

• The original statement “for all nonzero integers $a$ and $b,$ there exist integers $s$ and $t$ such that $\gcd(a,b)=as+bt$” can be translated as $$\forall\, a,b\bigg(a,b\in\mathbb Z\setminus\{0\}\implies\exists\, s,t\in\mathbb Z \;\gcd(a,b)=as+bt\bigg).\tag1$$
• Therefore, its converse is $$\forall\, a,b\bigg(\exists\, s,t\in\mathbb Z \;\bigg[\gcd(a,b)=as+bt\bigg]\implies a,b\in\mathbb Z\setminus\{0\}\bigg),\tag2$$ which translate to natural language as “for all variables $a$ and $b,$ if there exist integers $s$ and $t$ such that $\gcd(a,b)=as+bt,$ then $a$ and $b$ are nonzero integers.”

On the other hand, your suggestion “for all nonzero integers $a$ and $b,$ if there exists integers $s$ and $t$ such that $as+bt=d,$ then $d=\gcd(a,b)$” translates as $$\forall \,d\;\;\forall\, a,b\in\mathbb Z\setminus\{0\} \bigg(\exists\, s,t\in\mathbb Z \:\:\bigg[d=as+bt\bigg]\implies d=\gcd(a,b)\bigg),\tag3$$ which is not actually the converse of the original statement.

P.S. None of this is discussing validity or actual truth—just about translation between first-order logic and natural language.

P.P.S. Statements $(1),(2),(3)$ are logically equivalent to $$\forall\, a,b\;\:\exists\, s,t\bigg(s,t\in\mathbb Z \:\text{ and } \bigg[a,b\in\mathbb Z\setminus\{0\}\implies\gcd(a,b)=as+bt\bigg]\bigg),\tag4$$ $$\forall\, a,b\;\;\forall\, s,t \bigg(\bigg[s,t\in\mathbb Z \:\text{ and } \gcd(a,b)=as+bt\bigg]\implies a,b\in\mathbb Z\setminus\{0\}\bigg),\tag5$$ $$\forall \,d\;\;\forall\, a,b \;\;\forall\, s,t \bigg(\bigg[a,b\in\mathbb Z\setminus\{0\} \:\text{ and }\: s,t\in\mathbb Z \:\text{ and }\: d=as+bt\bigg]$$$$\implies d=\gcd(a,b)\bigg),\tag6$$ respectively.