Give an example of a set which is not transitive

Question

Transitive set: set $x$ is transitive if $\forall y\in x(y\subseteq x)$

I think $\{\varnothing\}$ is not transitive since $\varnothing\in\{\varnothing\}$ but $\varnothing\not\subseteq\{\varnothing\}$

Can someone verify this please.

Thanks in advance for the help

Answer 1

That doesn't work. The empty set is a subset of any set, so in particular $\varnothing\subseteq \{\varnothing\}$.

It's not a bad idea to think of a singleton, just not that one ...

Answer 2

You can show that, assuming $\sf ZF$, every transitive set is either $\varnothing$, or that $\varnothing$ is an element of that set.

Take any non-empty set which does not have the empty set as an element.

For example $\mathcal P(X)\setminus\{\varnothing\}$, when $X\neq\varnothing$, is such example. Other examples include $\{A\subseteq X\mid |A|\geq\aleph_0\}$, where $X$ is an infinite set, or any set of singletons etc.