1. ## Extensionality and well-founded

Let $\displaystyle \vartriangleleft$ and $\displaystyle \hat{\vartriangleleft}$ be relations on sets $\displaystyle X$ and $\displaystyle \hat{X}$ respectively.

The structure $\displaystyle (X, \vartriangleleft)$ is:

1). extensional if $\displaystyle \forall x \in X \forall y \in X [\forall z \in X(z \vartriangleleft x \leftrightarrow z \vartriangle y) \rightarrow x=y]$,

2).well-founded if $\displaystyle \forall A[\phi \neq A \subset X \rightarrow \exists a \in A \forall x \in A ¬ (x \vartriangleleft a)]$

Show that if $\displaystyle (X, \vartriangleleft)$ and $\displaystyle (\hat{X}, \hat{\vartriangleleft})$ are extensional and well-founded, then there is at most one isomorphism between them.

Hint: Consider $\displaystyle \{ x \in X | f(x) \neq x \}$ for an automorphism $\displaystyle f$ of $\displaystyle (X, \vartriangleleft)$.

I have absolutely no idea how to do this question. I can't see how the hint is helpful. If I consider the automorphism, i'm not considering $\displaystyle (\hat{X}, \hat{\vartriangleleft})$ which I need to do.

Can anyone point me in the right direction?

2. Well an isomorphism has an inverse. So if you have two isomorphisms $\displaystyle g$ and $\displaystyle h$, you can consider the automorphism $\displaystyle h^{-1}\dot g$.

If the two automorphisms are distinct, then the composition I suggested is an automorphism which is not the identity. So it suffices to show that the only automorphism of X is the identity map. Use the hint to show this.