How to define a set without set builder notation

[hgfhh12]

How do you define a set without using set builder notation? For example, let's say that I want to define set S as:

S={x ∈ ℕ ∣ 0<x<5}

Then
S={1,2,3,4}
However, suppose that I wanted to define S without set-builder notation, as below?

∀x(x ∈ ℕ ^ 0<x<5 ⟺ x∈S )

Would these two statements be equivalent, or is there something else provided in the set builder notation that I am missing?

Thanks.

 

[micromass]

The two statements are completely equivalent.

 

[jambaugh]

Your definition is fine. In essence a set is isomorphic to the logical predicate defining which elements are in the set. Set algebra and boolean algebra are isomorphic. (This is true provided we forbid self reference in predicates which would allow formulation of a Russell's paradox.)

So for example {} = X such that for all a in X, True=False. (there is thus never an a in X).

We typically do the reverse however. We like to map logic into set notation and set concepts. See for example introductory probability theory.