$\bf Question$
Let $L=(F=\{f^2,g^2\}, P=\{=\}, C=\{c\})$ be a first order language.
Let $I = (\Bbb R^{2\times 2}, f^2(x,y)=(x+y)^t,g^2(x,y)=x+y,0) $ be an interpretation of $L$.
Exhibit a sentence $\alpha$ that has $I$ as a model but that is not universally valid. $\alpha$ must contain at least once each of $f,c, =$.
$\bf Attempt$
Let $\alpha= \exists x \exists y (\neg x=y \wedge f^2(x,y)=c)$.
Then $I$ is a model of $\alpha$ because it is true by taking $x=-y, y=I$.
Let $J=(\Bbb N ,f^2(x,y)=x \cdot y,g^2(x,y)= x \cdot y , 1)$
Then $\alpha$ is false because there doesn't exist $x,y, x\neq y$ such that $xy=1$.
Is this correct?