which will be the image of this function $f:A \to A, f(a)=1+(a*a)$?

Question

If I have a set with one element for example A={1} more info: the language is: L=(f,g,c). the L-structure is: (A, . ,+,1) the formula is: ∀xƎy(1+(x . x)=y) My textbook claims that every structure is ok with this formula but my doubt is if this is true even with a domain that have just an element.

Answer 1

IF the domain is $A = \{ 1 \}$, however we define the interpretation of the function symbols, they must be interpreted with binary functions $f^{A}, g^{A} : A \times A \to A$.

Thus, they are functions that for input $1$ will output $1$, i.e. such that $1 \cdot 1 = 1$ and $1+1 = 1$.

In addition, we have also $c^A=1$.

In conclusion:

$1 + (1 \cdot 1)=1$ will be true,

that means that: $\exists y \ (1+ (1 \cdot 1)=y)$ holds in $A$, and thus also $\forall x \ \exists y \ (1+ (x \cdot x)=y)$ holds in $A$, and finally also:

$\forall x \ \exists y \ (c+ (x \cdot x)=y)$ holds.

So, yes: "every structure [with domain $A$] satisfies this formula."