# Is there a left extensional relation which is not right extensional?

A binary relation $$R$$ on a set $$S$$ is defined to be left extensional if the following property holds, where $$x$$, $$y$$, and $$z$$ refer to elements in $$S$$: $$(\forall x)(\forall y)((\forall z)(zRx \leftrightarrow zRy) \rightarrow x=y)$$. Right extensional is the same thing, except with $$xRz$$ replacing $$zRx$$, and $$yRz$$ replacing $$zRy$$. My question is, what is an example of a binary relation that is left extensional but not right extensional?

Here is an example with a set (and not a proper class). Take $$S=\{a,b,c\}$$ with

• $$aRa$$, $$aRb$$,
• $$bRa$$, $$bRb$$,
• $$cRb$$, $$cRc$$.

The relation $$R$$ is left extentional since the three sets

• $$\{x\in S\mid xRa\} = \{a,b\},$$
• $$\{x\in S\mid xRb\} = \{a,b,c\},$$
• $$\{x\in S\mid xRc\} = \{c\}$$

are distinct.

It is not right extensional since

$$\{x\in S\mid aRx\} = \{x\in S\mid bRx\} = \{a,b\}.$$

There does not seem to be an example if $$S$$ has less than three elements.

Sure. Consider a model of $$\mathsf{ZF-Extensionality}$$ with at least one urelement. $$\mathsf{Pairing}$$ ensures upwards (= left) extensionality with respect to $$\in$$, but the existence of urelements kills downwards (= right) extensionality with respect to $$\in$$.

• Nice example! But I think left and right should be the other way around(?) Commented Apr 11 at 23:45