Consider the signature with just one function symbol, $\times$, and constants 0 and 1.

Question

Consider the signature with just one function symbol, $\times$, and constants 0 and 1.

1. M1 is the usual integers $Z$ with the usual interpretations of $\times$, 0, and 1.
2. M2 is the usual real numbers $R$ with the usual interpretations of $\times$ , 0, and 1.
3. M3 is the usual complex numbers $C$ with the usual interpretations of $\times$, 0, and 1.

Question:

• Find a sentence $\sigma$ holds for M1 but not for M2 or M3
• Find a sentence $\gamma$ holds for M2 but not for M1 or M3
• Find a sentence $\phi$ holds for M3 but not for M1 or M2

I stuck with this question for a while. I really need a hint to start with. Thank you.

Answer 1

Hint: For any given, fixed, natural number $n$, what numbers are expressible as $a^n$? Does it change if $a$ is real, complex, or an integer?

Answer 2

A sentence that holds in $M_1$, but not in $M_2$ or $M_3$:

$$a{\,\times\,}b=1 \rightarrow a = b$$

A sentence that holds in $M_2$, but not in $M_1$ or $M_3$:

$$ {\large(} (a{\,\times\,}(a{\,\times\,}a)) = (b{\,\times\,}(b{\,\times\,}b){\large)} {\large)} \rightarrow a=b) \land {\large(} \forall a,\exists b\, ((b{\,\times\,}(b{\,\times\,}b))=a) {\large)} $$

A sentence that holds in $M_3$, but not in $M_1$ or $M_2$:

$$\forall a, \exists b\,{\large(}b{\,\times\,}b=a{\large)}$$