# How to define a set without set builder notation

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

$$S = \{ x \in \mathbb{N}\ \mid 0 < x < 5\}$$

Then $$S = \{1, 2, 3, 4\}$$

However, suppose that I wanted to define S without set-builder notation, as below?

$$\forall x \in \mathbb{N} (\ 0 < x < 5 \iff x \in S \ )$$

Are these two statements equivalent?

• Make that a $\iff$ instead of a $\implies$. – Jack M Nov 20 '13 at 23:00
• Come to think of it, even that doesn't really work, since it doesn't rule out the possibility that, for instance, $\pi\in S$. – Jack M Nov 20 '13 at 23:04

They are not equivalent. If $S'=\{1,2,3,4,5,6,7\}$ then we have : $$\forall x \in \Bbb N \quad 0<x<5 \Rightarrow x \in S'$$ but $S \neq S'$.

• The question has been edited to eliminate this counter-example. – AlexR Nov 20 '13 at 23:10

A fool-proof alternative would say $$\forall x: (x\in \mathbb N \wedge 0<x<5) \Leftrightarrow x\in S$$ This rules out all $x \notin \mathbb N$, which your initial proposal did not and it also formulates it as equivalencies.

Note that always the following representations of an arbitrary set $A$ are equivalent: $$A = \{x\in X: \phi(x)\}$$ and $$\forall x': (x'\in X \wedge \phi(x')) \Leftrightarrow x'\in A$$

Set-builder notation can be taken as a contextually defined abbreviation. At least this is how I learned it.

We can take $y\in\{x:\phi\}$ as short for $\exists z(y\in z \wedge \forall x(x\in z\leftrightarrow \phi))$. An obvious variation works for $\{x:\phi\}\in y$ and for contexts involving "$=$". In principle, we don't usually need anything in the language of set theory but the machinery of FOL and $\in$, so pretty much anything else is a handy abbreviation.

Apart from errors in logic, what you have is a proposition which happens to be true for your set $S$. This is not the same thing as an expression for $S$. You can't, for instance, write:

$$3\in \left(\forall x \in \mathbb{N} (\ 0 < x < 5 \Rightarrow x \in S \ )\right)$$

Whereas you can write $3\in \{1,2,3,4\}$.

• Okay, I understand that it should be "if and only if". Then, how do I define S without using this set builder notation? – Thomas Nov 20 '13 at 23:05
• He is trying to define a set. The definition is not the same thing with what is defined. So you never write $x\in Definition$. I don't think what you say is really a problem. – hhsaffar Nov 20 '13 at 23:06
• @hhsaffar Nice explanation ;-) – AlexR Nov 20 '13 at 23:09