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A variable in an equation may be replaced by any of the numbers in its domain. The resulting equation may be either true or false.

Here is another way to show that the domain of a variable $y$ is $\lbrace$$0, 1, 2, 3$$\rbrace$:

$$y\in\lbrace 0, 1, 2, 3\rbrace$$

(Read $"y$ $\color\red{\text{belongs to}}$ the set whose members are $0, 1, 2, 3"$.)

Replacing each variable in an an open sentence by each of the values in its domain is a way to find solutions of the open sentence.

Question: To me, the red typed words: $"\color\red{\text{belongs to}}"$ says the variable belongs to the set; while the the other two statements, which use the possessive $\text{"its domain"}$, says the set belongs to the variable. Is there a non-contradictory interpretation that I am missing?

Would "is a member of" be a such a non-contradictory interpretation and reading of the symbol? Then the statement would read: "$y$ is a member of the set whose members are $0, 1, 2, 3$". Which seems kind of wordy.

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  • $\begingroup$ "Its domain" relates to the "open sentence", not to the set. A set does not have a domain, a domain is a set. $\endgroup$ – Yves Daoust Sep 18 '14 at 7:48
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    $\begingroup$ no "its domain" relates to the variable, the domain is the set of values that the variable can take. $\endgroup$ – Denis Sep 18 '14 at 9:36
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    $\begingroup$ its about as contradictory as the statement "I am a $member$ of $my$ family." $\endgroup$ – John Joy Sep 18 '14 at 13:39
  • $\begingroup$ Simply read it as "what y can be replaced by belongs to" $\endgroup$ – skullpetrol Feb 13 '15 at 17:21
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As for "$\in$" see the other answers. Concerning the idea of a variable:

Each letter, say $y$, denoting a variable comes a priori with its domain $D_y$, a certain set. We are allowed to replace this $y$ in the formula by any element $a\in D_y$ and obtain a proposition about constants which is either true or false.

See also here:

High school math definition of a variable: the first step from the concrete into the abstract...

and here:

Is the 'variable' in 'let $y=f(x)$' free, bound, or neither?

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  • $\begingroup$ Thank you for the helpful links. $\endgroup$ – skullpatrol Sep 19 '14 at 17:03
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The relation $\in$ is typed \in (in TeX) and often pronounced "is in", or "is a member of", or "is an element of". But many variations occur occuring to the particular set, so you might pronounce $z\in\mathbf C$ as "$z$ is a complex number" rather than "$z$ is an element of the set of complex numbers".

Your confusion is due to the two different roles of a variable. A variable is a name, a lexical object introduced in a mathematical text, usually specifying what kind of value it designates by specifying a set of allowed values, its domain. The domain is an attribute of the name of the variable. However, when reasoning about the variable, one assumes that it designates one specific element of its domain, the (current) value of the variable. When talking about $y$ in this way, one always means the value of (the name) $y$, not the name $y$ itself. It is the value of $y$ that is an element of (is a member of, is in) the domain of (the name) $y$.

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  • $\begingroup$ Would these two different roles of a variable be more clearly shown in the phrases: "any value of the variable in its domain" versus "the specific, but as of yet unknown, value of the variable"? $\endgroup$ – skullpatrol Sep 26 '14 at 10:51
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"You belong to me", "I belong to you"... Possession always causes confusion.

A set, by definition, is a collection of elements. A given element $x$ can either be in a given collection, or not in the collection. I read "$y\in\{0,1,2,3\}$" as "$y$ is one of the elements $0,1,2,3$". No belonging involved.

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The most general ways to "pronounce" $\in$ certainly are "is an element of" or "is a member of". However, in a case like this one where the set is not only finite but also very small it might make sense to read "$y \in \{1,2,3\}$" as "$y$ is either $1$ or $2$ or $3$" or "$y$ is one of the values $1$, $2$, and $3$".

This is in accordance with, say, reading "$y \in \mathbb{N}$" as "$y$ is a natural number".

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