Transitive sets

Statement

A set $A$ is transitive if every element of $A$ is also a subset of $A$, i.e. $\forall x\,(x \in A \to x \subseteq A)$. Decide which of the following sets are transitive.

(i) $\varnothing$ (ii) $\{\varnothing\}$ (iii) $\{\{\varnothing\}\}$ (iv) $\{\varnothing,\{\varnothing\}\}$

Solution

(i) Transitive: $\varnothing$ has no elements, so the condition holds vacuously. (ii) Transitive: the only element is $\varnothing$, and $\varnothing \subseteq \{\varnothing\}$. (iii) Not transitive: the only element is $\{\varnothing\}$; to be a subset we would need $\varnothing \in \{\{\varnothing\}\}$, which is false (the only element is $\{\varnothing\}$, not $\varnothing$). (iv) Transitive: the elements are $\varnothing$ (a subset of every set) and $\{\varnothing\}$, and $\{\varnothing\} \subseteq \{\varnothing,\{\varnothing\}\}$ because $\varnothing$ belongs to the set.