Expressing the instance of a ZF Set Theory axiom for a given property

Question

I am currently in the process of studying ZF Set Theory (without the Axiom of Choice) and I have come across a type of question that is unclear to me. The basic format of the question is to "express the instance of a given axiom for a particular property".

For example an example of this type of question would be:

Without using abbreviations, only using logical symbols and the relation symbols =, ∈ only). Express:

• the instance of the Axiom Schema of Separation for the property “$x$ is not a singleton”

• the instance of the Axiom Schema of Separation for the property “$x$ is transitive”

Now of course I am familiar with the properties of a set being a singleton or a set being transitive. Equally, I am also familiar with the Axiom Schema of Separation. However, what I'm not sure is exactly what this type of question actually means. I can certainly state the formal definitions of these properties and the Axiom of Separation - but it seems to suggest that I am supposed to deduce some relationship between the two.

What is the question really asking and how do I answer this type of question?

Answer 1

I'll do the first one.

Axiom Schema of Seperation: For y not free in the formula φ:

$$ ∀x∃y∀z(z∈y ↔ (z∈x) \land φ) $$

And similarly, we also have:

"z is not a singleton": $$∀a(\{ a \} ≠ z) $$

Now let $\phi = \forall a (\{a \} \neq z)$. Where z ≠ y. So, the instance of the Axiom Schema with our φ is:

$$∀x∃y∀z(z∈y ↔ z∈x \land ∀a(\{a\} ≠ z)) $$

In English, for any set $x$ and $y ≠ z$, (otherwise $y$ would be a free variable), we have that $y = \{z∈x | ∀a(\{a\} ≠ z)\}$ is a set.

(i.e. $y ⊆ x$ and the elements of $y$ are not singletons).