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As a programmer learning basic set theory, I'm a bit confused by the symbol $\in$.

For example, let $a$ be a real number and an element of a set $A$. What is the "value" of the expression $a\in A$? Is it

  1. a truth value, such that $(a\in A)\in \{false, true\}$, or
  2. the number $a$, or
  3. either of the above, depending on the context?

Also, given another set $B$ such that $A\in B$, is the expression $a\in A\in B$ valid, and is it supposed to be read as $(a\in A)\land(A\in B)$?

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    I just edited the title to make it more grammatically pleasing. It's still kind of unclear, and I was going to add MathJax, but it's on the Hot Questions list and I think that would bump it off. – 2017-02-19
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    Maybe it's just me, but I think the title "Is the 'element of' relation a logical statement with a truth value?" would have more clearly represented (to me) what the question is asking. – 2017-02-19
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    Unlike programming, mathematics is full of abuses of notation. Things that would be technically inconsistent from a programming point of view are used all the time where the meaning is clear to a human with context. – 2017-02-19
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    @asmeurer Programming is not completely absent of abuses of notation. In Perl you can essentially use the same name for two different variables if you treat it as both an ordinary variable and a reference. It's not something you should ever do and it can cause bugs that are very hard to find, but it is possible. – 2017-02-19
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    @MattSamuel Perl's grammar also isn't context free. But more...sane programming languages are. – 2017-02-19
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    @MattSamuel: Possibly splitting hairs here, but how does "the" expression make sense in the title? Shouldn't it either be "the" relation or "an" expression? (By expression I meant the "element of" symbol _and_ the left and right-hand objects, all as a whole.) – 2017-02-21
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    @Simplex Sure, changed it to "an." – 2017-02-21

4 Answers 4

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The value of $a \in A$ is always a truth value.

However, there are grammatical contexts where it doesn't make sense to ask about the value of a string of symbols. For example, in the phrase

For every $x \in\mathbb{R}$, $x^2 \geq 0 $

the phrase $x \in \mathbb{R}$ is not used in a context where one would speak it having a value; instead the expression is used to do something different:

  • Introduce a new variable $x$
  • Bind $x$ to the type of real numbers
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    From a programmer perspective, one could say that $\in$ can be used not only as a predicate, but also as a "type ascription/assertion" operator, as it is common in functional languages (eg. in Haskell, you could write `x = y :: Int`, meaning that "`y` must be an integer in this term"). This is quite coherent with the intuitive interpretation of types as sets, although there one should be careful for the same reasons @gardenhead notes, not confusing "object sets" and "meta sets". – 2017-02-19
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The expression $a\in A$ is a logical statement: it is the assertion that $a$ is an element of $A$. So its "value" is a truth value: either $a$ is an element of $A$ and the statement is true, or $a$ is not an element of $A$ and the statement is false. In most contexts we don't think of logical statements as having "values" though; we just say whether they are true or false (and usually only state them if they are true!).

There are some contexts, however, where the notation $a\in A$ is used with the "value" $a$. For instance, if you say "There exists $a\in A$ such that..." then you are asserting the existence of $a$, not the existence of the truth value of $a\in A$. More specifically, you are asserting the existence of $a$ such that in addition, the statement $a\in A$ is true. Another similar context is in set-builder notation: $$\{a\in A:a^2=a\}$$ refers to a set of values of the variable $a$ (such that $a\in A$ is true and $a^2=a$ is true), not a set of truth values of $a\in A$.

Note that these usages are really just quirks of the grammar of mathematical writing, and are not at all particular to the symbol $\in$. For instance, you can also write "There exists $x>0$ such that..." where you are referring to the existence of $x$ such that $x>0$ is true, not the existence of the truth value of the statement $x>0$.

The notation $a\in A\in B$ is not commonly used, though presumably it would mean "$a\in A$ and $A\in B$". In general, we only "chain together" statements like this when we are talking about a transitive relation. So for instance, we write $xdo commonly chain together $a\in A\subseteq B$, since these relations are "transitive" in that $a\in A$ and $A\subseteq B$ together imply $a\in B$.)

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    That note about transitive relations is really helpful, thanks. – 2017-02-18
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For your first question: The correct option is 3, i.e., it depends on context. For example, you can say

The number $5\in\mathbb{R}$ is a very nice number, and its name in English is "five'.

and you can also say

The statement $\sqrt{2}\in\mathbb{Q}$ is false, as was proven by the Greeks.

For your second question: Yes, that is how it should be read, $a\in A\in B$ means that $a\in A$ and $A\in B$. It's a similar sort of combined notation to saying $x\leq y\leq z$ for numbers, for example.

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    I would say that the interpretation of that first sentence is "the number 5, which is an element of $\Bbb R$, is ...". That is, the $5 \in \Bbb R$ is still a logical statement, and so has a boolean value, Though in this case there is an assertion that the value is true. – 2017-02-18
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    I abominate the first version. The number *$\text{mathematical statement}$* is a very nice number. What? – 2017-02-19
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    @GitGud But any alternative I've seen is needlessly verbose: "The number $5$, $5\in\mathbb{R}$, is a very nice number." Ugh! – 2017-02-19
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    @OliverFord The real number 5 is very nice. – 2017-02-19
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    @GitGud That's extremely common. That of course doesn't mean you have to like it, but I think it's fine. – 2017-02-19
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    @MarkS. I'm not sure if you're joking, but it's hopefully obvious that we're talking about the more general case of $x\in{Y\setminus{Z}}$, for example, as opposed to $x$, $x\in{Y\setminus{Z}}$. (The latter reads to me like two variables in the set, which I've accidentally given the same name $x$.) – 2017-02-19
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The other answers give the usual interpretation in informal mathematical reasoning, but there is also the point of view of formal mathematical logic.

In formal logic, specifically first-order logic, $a \in A$ is a proposition (not an expression). A proposition is a formal mathematical object, but it is not a member of any set (more accurately, it is not a member of any set within the theory itself). Now, a proposition may be true (or false), if we can prove it to be, but that does not mean it is equivalent to some object in our theory called true. The truthiness of a proposition is given by a meta-theoretic judgement $\phi \text{ true}$.