Talk:Element (mathematics)

Unicode?

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

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

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)

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)

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)

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

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)

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)

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)