# Math Help - Extensionality and well-founded

1. ## Extensionality and well-founded

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

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

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

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

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

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

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 $(\hat{X}, \hat{\vartriangleleft})$ 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 $g$ and $h$, you can consider the automorphism $h^{-1}\dot g$.

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.