Talk:Element (mathematics)

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Unicode?

Is there a Unicode for "is element of" and "is not element of"? --137.193.16.78 19:57, 30 August 2007 (UTC)[reply]

Yes, U+2208 is ∈, U+2209 is ∉, U+220B is ∋, and U+220C is ∌. - MTC 12:14, 8 October 2007 (UTC)[reply]

Question...

... can you have the following set?

D = {2,4,6,8,10,12,2}

?? - Ta bu shi da yu 15:17, 10 November 2007 (UTC)[reply]

When I first encountered set theory I found this terribly confusing. Set theory writers such as Suppes and Halmos just introduce the notion "set" as: "That which is defined by its membership". Suppes notes that in completely axiomatic systems the principal primitive (i.e. undefined) notion is that of belonging (Suppes p. 2). So why can't we have a set comprised of 6 objects as follows:

Basket B is the set that consists of a basket with the following 6 fruits: { banana_1, banana_2, orange_1, lemon_1, orange2, orange_3 }

Note: set B always contains the empty basket symbolized here as { } or as ∅

{ banana_1, banana_2, orange_1, lemon_1, orange2, orange_3, ∅ }

We can. Perhaps each individual banana and orange has a number written on it. Certainly each one is a distinct object. Here the notion of membership [denotes] objects as truly separate objects, not descriptions of (concepts of) object-attributes.

If we want to move beyond the simplest formulation of "set as a collection of members", we have to employ the axiom of specification to create one or more subsets from the über-set that we started with (e.g. the basket of fruit). This is the Aussonderonnderaxiom (the "separating" axiom that pulls out a subset from the über-set (cf Suppes p. 6) "it permits us to separate off the elements of a given set which satisfy some property and form the set consisting of just those elements."(Suppes 1972:6):

"Axiom of specification. To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds" (Halmos 1960:6)

∉∈∸→⊂∀∃

"The precise form of the axiom is: (∃Y)(∀x)[x ∈ Y ←→ x ∈ Z & φ(x)]" (Suppes p. 7, typography changed to capital Y and Z to emphasize their set-hood).

To say that "There exists at least one set Y such that for any element x contained in set Y" is logically equivalent to saying that "For all x, where x an element of über-set Z AND the specification φ(x) is satisfied, x is an element of at least one set Y."

As noted by Suppes, the fact that an uber-set Z must be specified up front prevents Russel's paradox.

The most common notion "set of" thus has behind it the notion of an uber-set and the notion of "separation into subsets", that is, "class(ification) of", that is, a discription of one or more attributes. Once we have passed through the specification, the set will not be individual objects themselves, but rather representives of the prescribed attribute (i.e. "concept of" versus "instance of".) Thus a set can defined by description of how to find its members, assuming there are any "members" that meet the description. So: it so happens that a set (collection) does exist such that its definition = { membership } are: " 'A type of orange-colored citrus fruit' OR 'A type of yellow-colored citrus fruit' OR 'a type of yellow-colored tropical fruit'" (here "=" stands for "is identical to"

{ x ∈ B: those objects in the basket B that are fruit } = { orange, lemon, banana }.

{ x ∈ B: those fruit in the basket B that are yellow { lemon, banana }

The problem with the nubmers-example is that the numbers 12, here indicated by the symbol "12", are in set theory really sets defined by the specification axiomn, and thus abstract notions called "12". And, by the axiom of unions, the first set "12" can absorb the second set 12 because the two sets of ordered pairs are identical because they contain the same elements:

From these two sets {2, 4, 6, 8, 10, 12} ∪ { 12 } we can select elements such that we create the set { 2, 4, 6, 8, 10, 12 }.

And your example would be: "All the even but non-negative, non-zero integers up to and including 12" = { 2, 4, 6, 8, 10, 12 }.

Note: for this to work, all these terms and symbols -- even, non-zero, integer, 2, 4, 6, 8, 10, 12 -- must be defined somewhere beforehand.

Thus we say that, "The number '2' OR the number '2' OR even the number '2 OR, maybe ... the number '2' when added to the number '1' produces the number '3'" is logically equivalent to our saying: "The number '2' added to the number '1' produces the number '3' ". The number '2' is a "concept" here, not the instance of an object; '2'-ness is an attribute of a pair of anything". Thus: "Napoleon OR Napoleon OR maybe even Napoleon was exiled to Elba" means "Napoleon was exiled to Elba."

Both the above and the following requires the "axiom of extension" (two sets with the same members are identical if and only if (logically equivalent) they have the same members", but also the "axiom of specification" (stated above as how to define a set) and the "axiom of union" (this notion of absorbing or spawning set members).

The notion of "absorbing" members is how we combine one two sets into a single set, but it can turned around to "spawn" members so we can generate more sets from a given set. That is: { 2 } ←→ { 2 } ∪ { 2 }, where ←→ means "logically equivalent". And { 2 } ←→ { 2 } ∪ { 2 } ←→ { 2 } ∪ { 2 } ∪ { 2 }, etc.

{2, 4, 6, 8, 10, 12} ←→ { 2, 2, 4, 6, 8, 10, 12 } ←→ { 2, 2, 4, 4, 6, 8, 10, 12 } ←→ { 2, 2, 4, 4, 6, 8, 8, 10, 12 } ←→

{ 2, 4, 6, 8, 10, 12 } ∪ { 2, 4, 8 } i.e. the union of two sets, each with their own specification

And this union of two sets can be written as e.g. four unions by a similar method:

{ 2, 4, 6, 8, 10, 12 } ∪ { 2, 4, 8 } ∪ { 6, 12 } ∪ { 10 }

{ "Integers between 2 and 12, inclusive, that are even" } ∪ { "Integers between 2 and 12, inclusive, that are powers of two" } ∪ { "Integers between 2 and 12, inclusive, that have '3' as a factor AND are even" } ∪ { "Integers between 2 and 12, inclusive, that have '5' as a factor AND are even }

Bill Wvbailey (talk) 22:22, 11 December 2007 (UTC)[reply]

Membership in multisets

What about membership in multisets? For a particular element (or not-element) it can be true or false, but the membership can also be given a quantity, as the multiset article says:

A member of a multiset can have more than one membership. --Abdull (talk) 16:55, 2 February 2008 (UTC)[reply]

1∈2? Is this right? 85.157.127.160 (talk) 18:33, 12 May 2008 (UTC)[reply]

Origin

Where does this symbol meaning "is a member of", which looks a bit like the sign for the Euro currency, come from? Does it have a name? — Preceding unsigned comment added by 86.132.16.81 (talk) 16:29, 14 April 2013 (UTC)[reply]

It's a stylised lower case epsilon. It was first used for set membership by Peano who chose it because it is the first letter of the latin word est, meaning "is". 176.253.168.123 (talk) 22:22, 24 November 2013 (UTC)[reply]

Really lousy definition

An element is distinct, is the statement. So {1,1,1,2,2,3,3,3,3} isn't a set???? Or how about {1,2} and {2,1}: same set or different set?? Seems to me that (unless the set is ordered) the answer to my second question is "same" and the answer to my first question is "it depends". the set which contains the integer which is one greater than zero and the integer which is one less than two has one element, while the values of two observations may indeed form the set {1,1} (with some implicit measurement unit). This article does an absolutely miserable job of defining (or explaining, if you're of the school that believes some concepts are fundamental and can't be further reduced) what it is about.173.189.78.173 (talk) 05:44, 22 April 2015 (UTC)[reply]

Set of colours

"The elements of a set can be anything. For example, ${\displaystyle C=\{\mathrm {\color {red}red} ,\mathrm {\color {green}green} ,\mathrm {\color {blue}blue} \}}$, is the set whose elements are the colors red, green and blue."

What is this special formatting supposed to indicate? For me it draws attention to the ambiguity of convention; elements of set C appear to be three words, not three colours. Or is it a commonly understood mathematical convention that only the name of a colour formatted in that colour represents the concept of that colour? In that case, what does the word "red" formatted in normal black text represent? In short, I would really find more plainly put if this fragment read

"For example, ${\displaystyle C=\{\mathrm {red} ,\mathrm {green} ,\mathrm {blue} \}}$, is the set whose elements are the colors red, green, and blue."

(The normal way these words are used), or if we're worried readers won't know that the word red refers to the colour red, etc., we can link each word to the appropriate article about the colour. Or if we truly want to drive home the point that any and all elements are admissible, why not pick a truly disparate set?

For example, ${\displaystyle M=\{\mathrm {red} ,\mathrm {2} ,\mathrm {whale} \}}$, is the set whose elements are the colour red, the number 2, and the animal whale."

Orbis 3 (talk) 08:27, 19 November 2017 (UTC)[reply]

I agree. Paul August ☎ 17:06, 19 November 2017 (UTC)[reply]

"⋲" listed at Redirects for discussion

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"⋳" listed at Redirects for discussion

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"⋴" listed at Redirects for discussion

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"⋵" listed at Redirects for discussion

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Elements

How come there is no discussion here, in this article, of the word "elements" as in the title of the book by Euclid, or Euclid's Elements, or Elements? — Preceding unsigned comment added by 71.183.110.20 (talk) 05:54, 22 November 2022 (UTC)[reply]

Requested move 5 September 2025

It has been proposed in this section that Element (mathematics) be renamed and moved to Membership (mathematics). A bot will list this discussion on the requested moves current discussions subpage within an hour of this tag being placed. The discussion may be closed 7 days after being opened, if consensus has been reached (see the closing instructions). Please base arguments on article title policy, and keep discussion succinct and civil. Please use {{subst:requested move}}. Do not use {{requested move/dated}} directly. Links: current log • target log • direct move

Element (mathematics) → Membership (mathematics) – The primary topic of this should be the membership relation ⁠${\displaystyle \in }$⁠, not the state of being an element of a set. Aside from the lead, the content in this article is already primarily focused on this anyway, and both Set membership and ∈ already redirect here. – Farkle Griffen (talk) 16:26, 5 September 2025 (UTC)[reply]

oppose This article is about both and 'is an element of' is a more common name for the relation anyway—blindlynx 21:16, 5 September 2025 (UTC)[reply]

@Blindlynx, Why should this article be about both in the first place? They're fundamentally distinct topics, and if the title is kept, the relation would still deserve its own article since there's a lot to say about it. There's really almost nothing that can be said about being an element of a set, as literally everything can be an element of a set. – Farkle Griffen (talk) 21:45, 5 September 2025 (UTC)[reply]

I don't follow, being an element isn't the same as potentially being an element. Further, the quality of being an element is defined with the relation and vice verse, how are they fundamentally different?—blindlynx 21:53, 5 September 2025 (UTC)[reply]

One is a relation, the other is an object. Relations have properties, you can talk about relations formally, and relations are studied in math. The relation ${\displaystyle \in }$ is the foundational concept of math, and can be used to define all mathematical properties (Number, function, equality, isomorphism, etc). The relation really deserves its own article.

"An element of a set" is not a thing that can be formally studied on its own, and is only defined by the relation. But it is not the same in reverse. In fact, strictly speaking, "being an element of a set" is not defined by the axioms of ZFC at all—only the relation ${\displaystyle \in }$ is defined, and even then, it does not have to mean "set membership" or be given any semantic meaning at all. It can be studied purely formally in terms of its properties. – Farkle Griffen (talk) 22:15, 5 September 2025 (UTC)[reply]

"I don't follow, being an element isn't the same as potentially being an element."

@Blindlynx, Also, this line really doesn't make sense to me. In math, these are the same thing: there is no such thing as a "potential set." – Farkle Griffen (talk) 22:22, 5 September 2025 (UTC)[reply]

I'm open to changing the name to something else, the point is to change the primary topic from "Element of a set" to the membership relation.

But, just to be clear, nearly every source I can find explicitly calls this relation "membership". I can't find any sources that explicitly say the name of that relation is "is an element of". – Farkle Griffen (talk) 21:54, 5 September 2025 (UTC)[reply]

For a generic relation, we use one symbol for both the relation as a thing (R) and an instance of the relation (xRy) and there is no question about which of these is "the name" of the relation. For the relation of ${\displaystyle \in }$, if I were referring to it by itself I would call it "set membership", but each instance of it (and perhaps also the name of the symbol) is pronounced "is an element of". To me this feels more an issue of grammar than of substance. ----JBL (talk) 19:26, 7 September 2025 (UTC)[reply]

oppose. I think element is the more common term. No objection to a redirect from the proposed, or more likely from Member (mathematics). Russ Woodroofe (talk) 07:18, 6 September 2025 (UTC)[reply]

I can get behind the Element of a set proposal below! Russ Woodroofe (talk) 08:02, 15 September 2025 (UTC)[reply]

Support move to either Element of a set or Set membership. I don't care very much about the semantic distinction here, but there's no reason for this parenthetical title structure when natural disambiguation is possible. ----JBL (talk) 19:26, 7 September 2025 (UTC)[reply]

• Support Element of a set as WP:NATDIS. ~ Jenson (SilverLocust 💬) 04:09, 15 September 2025 (UTC)[reply]