In Chapter 6 Exercise 9 (p.145) of Introduction to Logic, Tarski asks the reader to prove that the following statements, taken as axioms, are independent:
Theorem III. For any elements $x$, $y$, and $z$ of the set $S$, if $x\cong y$ and $x\cong z$, then $y\cong z$.
Theorem IV. For any elements $x$, $y$, and $z$ of the set $S$, if $x\cong y$ and $y\cong z$, then $z\cong x$.
Theorem V. For any elements $x$, $y$, $z$, and $t$ of the set $S$, if $x\cong y$, $y\cong z$, and $z\cong t$, then $x\cong t$.
No other axioms appear to be assumed in the question, and Tarski seems to be using the $\cong $ symbol here to designate any relation whatsoever, not necessary an equivalence relation (for an example of this usage, see pp.121-124).
My problem: I have not yet found a model where (III) and (IV) hold while (V) fails. In fact, I seem to be able to prove that (III) and (IV) imply (V). Here is my argument (Edit: thanks to @DanielV for providing a much simpler argument, reproduced here):
Suppose $x\cong y\cong z\cong t$. Then $z\cong x$ follows from (IV). By (III), $z\cong x$ and $z \cong t$ give us $x\cong t$, as needed. $\square$
My question: What am I missing? Assuming I am wrong, I would appreciate learning the error in my argument and, ideally, seeing an example of a model in which (III) and (IV) hold but (V) fails.
IMO, the problem is stated diffrently
Show that the three following "axioms" are independent: Axiom I and Theorems I and II of Sect.37, that are: