# Is this a valid set definition?

Is this set definition valid:

$$A = \{ |x| < 4 : x\text{ is an element of }\mathbb Z\}\quad?$$

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Are you sure you don't mean $A=\{-3,-2,-1,0,1,2,3\}$? –  Daan Michiels May 17 '12 at 15:44
But the definition is in inverse order, I mean all the objects of the form |x| < 4. –  mehdi May 17 '12 at 15:45
To show part of the problem with your definition, one relatively legitimate way of reading it would have it define the set {true, false}, since the 'evaluation' to the left of the colon could be said to return the result true for certain elements of $\mathbb{Z}$ (e.g., $x=-1$) and false for other elements of $\mathbb{Z}$ (e.g., $x=8$). –  Steven Stadnicki May 17 '12 at 16:32

No, that's not correct notation. If you mean the set of all elements $x$ of $\mathbb{Z}$ such that $|x|<4$, then the standard set-builder notation is $\{ x \in \mathbb{Z} : |x|<4 \}$. The name of a general element of the set ($x$) goes before the colon, along with the statement of the universe of discourse ($\mathbb{Z}$). The condition an element of the universe has to satisfy to be in the set ($|x|<4$) goes after the colon.

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no I know what A = {-3 .. 3} is I intentionally ask the inverse one if it is correct all objects of the form |x| < 4 –  mehdi May 17 '12 at 15:49
A = {some object definition : condition to be satisfied} –  mehdi May 17 '12 at 15:50
@mehdi: I can't understand your comment at all. What set were you trying to define? What does it mean for an object to be "of the form $|x|<4$"? –  Chris Eagle May 17 '12 at 15:50
I wonder its meaning too thats why I asked if it is a valid definition. I am not trying to define any set just wonder is such def. valid –  mehdi May 17 '12 at 15:54
@mehdi: Huh? Chris just answered your questions that it isn't a valid definition as you wrote. (Manzooret chie daghighan!) –  Gigili May 17 '12 at 15:59

The general form of class builder notation is

$$\{ f(x) : x \in S \mid P(x) \}$$

where $f$ is some function expression, $S$ is a class, and $P$ is a unary predicate. If $S$ is a set, then this notation also denotes a set, and we call this 'set builder notation'. I've borrowed the : and | from the magma language.

For example, the set {0, 1, 4, 9} could be given by

$$\{ x^2 : x \in \mathbb{Z} \mid 0 \leq x \leq 3 \}$$

Usually one doesn't write all three parts. You can omit the first part:

$$\{ x \in \mathbb{Z} \mid 0 \leq x \leq 10 \} = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$$

You can omit the last part:

$$\{ \sin(\pi x) : x \in \mathbb{Z} \} = \{ 0 \}$$

You can omit the middle part (but should be careful about forming proper classes!):

$$\{ \{ x \} : \forall y, y \notin x \} = \{ \{ \emptyset \} \}$$

more commonly, you combine the second and third parts

$$\{ 2x \mid x \in \mathbb{Z} \wedge |x| < 2 \} = \{ -2, 0, 2 \}$$

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If you mean that $A = \{ \lvert x \lvert : \lvert x \lvert < 4, \text{ and } x \in \mathbb{Z}\}$, then indeed you have a valid set. In that case you have in fact that $$A = \{ 0, 1, 2, 3 \}.$$

However, the notation is a bit off as there might be a confusion about whether you mean that $A$ is a set where the elements are $\lvert x \lvert$ or if you mean that $A$ is the set where the elements are $x\in \mathbb{Z}$. In both cases, however, you actually have a set.

The only thing that someone might raise a finger at is the notation.

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the elements of set A are "∣x∣ < 4" not a single number that makes me confused |x| is a single number so no problem –  mehdi May 17 '12 at 15:52
Yes, from your notation I would understand $A$ as the set of elements $\lvert x \lvert$ where $x \in \mathbb{Z}$ and $\lvert x \lvert < 4$. Just to make sure: Are you asking whether you actually have a set, or are you asking if the notation is correct? –  Thomas May 17 '12 at 15:54
I don't think that anyone would see the notation as meaning that $A = \{-3, -2, ... , 2, 3\}$ by the way. If that is what you want, then the notation is definitely off. –  Thomas May 17 '12 at 15:55