Set

A set is an un­ordered col­lec­tion of dis­tinct ob­jects. The ob­jects in a set are typ­i­cally referred to as its el­e­ments. A set is usu­ally de­noted by list­ing all of its el­e­ments be­tween braces. For ex­am­ple, \(\{1, 3, 2\}\) de­notes a set of three num­bers, and be­cause sets are un­ordered, \(\{3, 2, 1\}\) de­notes the same set.\(\{1, 2, 2, 3, 3, 3\}\) does not de­note a set, be­cause the el­e­ments of a set must be dis­tinct.

Another way to de­note sets is the so-called ab­strac­tion method, in which the mem­bers of a set are given an ex­plicit de­scrip­tion, leav­ing no need for list­ing them. For ex­am­ple, the set from the ex­am­ple above \(\{1, 3, 2\}\) can be de­scribed as the set of all nat­u­ral num­bers \(x\) which are less than \(4\). For­mally that is de­noted as \(\{x \mid (x < 4) \text{ and } (x \text{ is a natural number})\}\).

I am new to us­ing La­tex for edit­ing, so I am not sure if this is the best way to dis­play what’s above. Also, “x is a nat­u­ral num­ber” is the same as x∈N but we did not get to the mem­ber­ship op­er­a­tor yet.

Us­ing the ab­strac­tion method al­lows for de­not­ing sets with in­finitely many el­e­ments, which would be im­pos­si­ble by list­ing them all. For ex­am­ple, the set \(\{x \mid x = 2n \text{ for some natural } n \}\) is the set of all even num­bers.

A set doesn’t need to con­tain things of the same type, nor does it need to con­tain things that can all be brought to the same place: We could define a set \(S\) that con­tains the ap­ple near­est you, the left shoe which you wore last, the num­ber 17, and Lon­don (though it’s not clear why you’d want to). Rather, a set is an ar­bi­trary bound­ary that we draw around some col­lec­tion of ob­jects, for our own pur­poses.

The use of sets is that we can ma­nipu­late rep­re­sen­ta­tions of sets and study re­la­tion­ships be­tween sets with­out con­cern for the ac­tual ob­jects in the sets. For ex­am­ple, we can say “there are 35 ways to choose three ob­jects from a set of seven ob­jects” re­gard­less of whether the ob­jects in the set are ap­ples, peo­ple, or ab­stract con­cepts. noteThis is an ex­am­ple of ab­stract­ing over the ob­jects.

It’s also worth not­ing that a set of el­e­ments is it­self a sin­gle dis­tinct ob­ject, differ­ent from the things it con­tains, and in fact, one set can con­tain other sets among its el­e­ments. For ex­am­ple, \(\{1,2,\{1,2\}\}\) is a set con­tain­ing three el­e­ments: \(1\), \(2\) and \(\{1,2\}\).

Examples

  • \(\{1,5,8,73\}\) is a set, con­tain­ing num­bers \(1\), \(5\), \(8\) and \(73\);

  • \(\{\{0,-3,8\}\}\) is a set, con­tain­ing one el­e­ment — the set \(\{0,-3,8\}\);

  • \(\{\text{Mercury}, \text{Venus}, \text{Earth}, \text{Mars} \}\) is a set of four planets;

  • \(\{x \mid x \text{ is a human, born on 01.01.2000} \}\) is a set all peo­ple whose age is the same as the cur­rent year num­ber;

  • \(\{\text{author's favorite mug}, \text{Arbital's main page}, 73, \text{the tallest man born in London}\}\) is a set of four seem­ingly ran­dom ob­jects.

Set membership

Main page: Set membership

Set mem­ber­ship can be stated us­ing the sym­bols \(∈\) and \(∉\). They de­scribe the con­tents of a set.\(∈\) in­di­cates what is in a set. \(∉\) in­di­cates what is not in a set. For ex­am­ple, “\(x ∈ A\)” trans­lated into English is ”\(x\) is a mem­ber of the set \(A\).” and “\(x ∉ A\)” trans­lates to ”\(x\) is not in the set \(A\).”

Set cardinality

Main page: Cardinality

The size of a set is called its car­di­nal­ity. If \(A\) is a finite set then the car­di­nal­ity of \(A\), de­noted \(|A|\), is the num­ber of el­e­ments \(A\) con­tains. When \(|A| = n\), we say that \(A\) is a set of car­di­nal­ity \(n\). There ex­ists a bi­jec­tion from any finite set of car­di­nal­ity \(n\) to the set \(\{0, ..., (n-1)\}\) con­tain­ing the first \(n\) nat­u­ral num­bers. We can gen­er­al­ize this idea to in­finite sets: we say that two in­finite sets have the same car­di­nal­ity if there ex­ists a bi­jec­tion be­tween them. Any set in bi­jec­tive cor­re­spon­dence with \(\mathbb N\) is called countably in­finite, while any in­finite set that is not in bi­jec­tive cor­re­spon­dence with \(\mathbb N\) is call un­countably in­finite. All countably in­finite sets have the same car­di­nal­ity, whereas there are mul­ti­ple dis­tinct un­countably in­finite car­di­nal­ities.

See also

Children:

  • Set builder notation
  • Cardinality

    The “size” of a set, or the “num­ber of el­e­ments” that it has.

  • Convex set

    A set that con­tains all line seg­ments be­tween points in the set

  • Operations in Set theory
  • Cantor-Schröder-Bernstein theorem

    This the­o­rem tells us that com­par­ing sizes of sets makes sense: if one set is smaller than an­other, and the other is smaller than the one, then they are the same size.

  • Disjoint union of sets

    One of the most ba­sic ways we have of join­ing two sets to­gether.

  • Empty set

    The empty set does what it says on the tin: it is the set which is empty.

  • Set product

    A fun­da­men­tal way of com­bin­ing sets is to take their product, mak­ing a set that con­tains all tu­ples of el­e­ments from the origi­nals.

  • Finite set

    A finite set is one which is not in­finite. Some of these are the least com­pli­cated sets.

  • Extensionality Axiom

    If two sets have ex­actly the same mem­bers, then they are equal

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.