# 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.

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.

• Right now in or­der to find this page (as op­posed to be­ing linked to it) I have to know that it’s called “set_math­e­mat­ics.” This seems in­el­e­gant and bad. I would rather the URL of this page be some­thing like /​p/​1lw/​set.

• @5hc: I’ve made the ap­pro­pri­ate changes to the markup to make text dis­play in MathJax (which is the LaTeX-syn­tax markup lan­guage used for maths on this site and on Stack Ex­change). How­ever, I think it’s a bug in Ar­bital (which I’ve just pointed out to the de­vel­op­ers) that it’s not ren­der­ing cor­rectly. (EDIT: I’ve al­tered the markup into a form that works around the bug. It just makes the markup look a bit less nice.)

In gen­eral, you can use \text{text here}; if you want to put maths in­line with the text here, you can use dol­lar signs:

 \text{Heinz $$57$$ Va­ri­eties}


Ad­di­tion­ally, if you have a math­e­mat­i­cal string you want to type­set, like “sin”, you can use \mathrm{sin} which shows up as $$\mathrm{sin}$$. (It so hap­pens that there is already a built-in sym­bol that does that for sin: \sin.)